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Brownian Motion, Markov Properties and Hitting Times

1 · Prerequisites

2 · Summary

Brownian motion is placed in its raw natural filtration Natural and usual augmented Brownian filtrations, which is then completed by the terminal null sets and made right-continuous. The four filtration conventions are kept distinct, and the germ sigma-algebra at zero The Brownian germ sigma-algebra at zero is the first consumer of the completion claims.

The transition operators The Brownian transition semigroup are shown to agree with the expectation $E [f (x + B_{t})]$ , to form a semigroup, and to have the Gaussian kernel representation The Brownian kernels form a semigroup. Conditioning a known state on independent noise Conditioning a known state and independent noise then yields the deterministic-time Markov identity Markov property of Brownian motion and the future-path Markov property Future-path Markov property, from which Blumenthal's zero-one law Blumenthal's zero-one law follows at the germ.

Continuous-time stopping times and their stopped sigma-algebras are fixed in Continuous-time stopping times and stopped sigma-algebras; closed-set hitting times are stopping times by the exact rational-distance formula Brownian closed-set hitting times are stopping times; and the strong Markov theorem Strong Markov property of Brownian motion restarts Brownian motion at an almost surely finite stopping time through dyadic ceilings. Reflection Brownian reflection principle then gives the maximum law Law of the Brownian maximum and the one-sided hitting-time distribution Distribution of a one-sided Brownian hitting time, hence almost-sure hitting of every level One-dimensional Brownian motion hits every point almost surely, recurrence One-dimensional Brownian motion is recurrent, and the two-sided exit probability Two-sided Brownian exit probability, whose shifted laws are fixed by Brownian motion started at x. The planar annular exit calculation Planar Brownian annular exit probability uses the same shifted laws and the discrete optional sampling theorem. The filtration conventions actually used are recorded in Raw versus usual filtrations in the strong Markov theorem.

Choice is declared wherever the Brownian, conditional-expectation or optional sampling interfaces require it, and the countable-choice use is identified at the distribution-function correspondence. The companion page brownian-motion-markov-properties-and-hitting-times-examples carries the concrete densities, crossing probabilities, exit computations, restart examples, and the boundary counterexamples.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Natural and usual augmented Brownian filtrations

Definition

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion on a probability space $(Ω, F, P)$ Brownian motion. First replace the ambient space by its completion, retaining the notation $(Ω, F, P)$ for this extension. This is supplied by Assuming countable choice, every measure space has a unique complete extension to its completion; AC supplies its countable-choice hypothesis by AC supplies countable selections and prescribed serial paths. The coordinate functions, their laws, and their raw sigma-algebras are unchanged. Four families are distinguished.

Raw natural filtration. $F_{t}^{0} : = σ (B_{s} : 0 \leq s \leq t)$ for $t \geq 0$ , and $F_{\infty}^{0} : = σ (⋃_{t \geq 0} F_{t}^{0}) = σ (B_{s} : s \geq 0)$ . Both sigma-algebras exist by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal, and $(F_{t}^{0})_{t \geq 0}$ is a continuous-time filtration in the sense of Continuous-time filtrations and all-pairs martingales, the smallest one to which $B$ is adapted. It contains no completion and no right-continuous augmentation.
Raw right limit. $F_{t +}^{0} : = ⋂_{u > t} F_{u}^{0}$ for $t \geq 0$ . At $t = 0$ this is the germ sigma-algebra used later on this page; the uncountable intersection is the decreasing intersection over the rational $u > t$ , since $t \mapsto F_{t}^{0}$ is increasing. No null sets are adjoined in this operation.
Ambient null ideal and completed raw filtration. Let $N : = {M \subseteq Ω : there is N_{0} \in F with P (N_{0}) = 0 and M \subseteq N_{0}}$ be the family of all subsets of ambient $P$ -null events. These sets are ambient-measurable because the ambient probability space was completed. This is a sigma-ideal: for a countable family choose null envelopes using AC, then take their union; subsets require the same envelope. Nullness follows from Null sets are closed under countable unions and, in a complete space, under arbitrary subsets. Put $\overline{F}_{t}^{0} : = σ (F_{t}^{0} \cup N), t \geq 0.$ Each $\overline{F}_{t}^{0}$ is a sub-sigma-algebra of $F$ , contains every member of $N$ , and $\overline{F}_{s}^{0} \subseteq \overline{F}_{t}^{0}$ for $s \leq t$ , so $(\overline{F}_{t}^{0})_{t \geq 0}$ is a filtration.
Usual augmentation. $F_{t} : = ⋂_{u > t} \overline{F}_{u}^{0}$ for $t \geq 0$ .

The following facts are part of the definition and are the form in which it is used later.

(a) $F_{t}^{0} \subseteq \overline{F}_{t}^{0} \subseteq F_{t}$ for every $t \geq 0$ , and every $F_{t}$ contains $N$ . (b) $(F_{t})_{t \geq 0}$ is increasing and right-continuous: for every $t \geq 0$ , $⋂_{s > t} F_{s} = F_{t} .$ Indeed, if $A \in ⋂_{s > t} F_{s}$ , fix $u > t$ and take $s = (t + u) / 2$ . Then $A \in F_{s} \subseteq \overline{F}_{u}^{0}$ . Since $u > t$ was arbitrary, $A \in F_{t}$ . Conversely $t \leq s$ gives $F_{t} \subseteq F_{s}$ , because $\overline{F}^{0}$ is increasing and the intersection defining $F_{t}$ ranges over the larger parameter set ${u > t} \supseteq {u > s}$ , so $F_{t} \subseteq ⋂_{s > t} F_{s}$ . (c) Every completed set differs from a raw set by a null set. Let $D_{t} : = {A \subseteq Ω : there is A_{0} \in F_{t}^{0} and M \in N with A △ A_{0} \subseteq M} .$ Then $D_{t}$ is a sigma-algebra containing $F_{t}^{0} \cup N$ : complementation preserves symmetric difference, and, after selecting countably many witnesses by AC, $(⋃_{k} A_{k}) △ (⋃_{k} A_{0, k}) \subseteq ⋃_{k} (A_{k} △ A_{0, k}) \subseteq ⋃_{k} M_{k} \in N .$ Thus $\overline{F}_{t}^{0} \subseteq D_{t}$ . Conversely, if $A △ A_{0} \subseteq M \in N$ , then $A △ A_{0} \in N$ , so $A = A_{0} △ (A △ A_{0}) \in \overline{F}_{t}^{0}$ . This proves equality, and also ambient measurability of every $A \in D_{t}$ . In particular, for $A \in \overline{F}_{t}^{0}$ one has $P (A) = P (A_{0})$ for such an $A_{0}$ , and $\int_{A} X d P = \int_{A_{0}} X d P$ for every integrable real or complex $X$ , by Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree applied to $X 1_{A}$ and $X 1_{A_{0}}$ . Apply it to indicators for the measure equality. If $P (A) = 0$ , its raw representative $A_{0}$ is null, so $A \subseteq A_{0} \cup M$ is contained in an ambient null envelope. Every subset of $A$ therefore belongs to $N$ . This proves completeness of each $\overline{F}_{t}^{0}$ . It also proves completeness of $F_{t}$ : a null member belongs to $\overline{F}_{t + 1}^{0}$ and hence all its subsets belong to $N \subseteq F_{t}$ . Thus every ambient null event and every one of its subsets belongs already to $F_{0}$ ; together with right-continuity, this is the usual-conditions convention used below. (d) The four families $F_{t}^{0}$ , $F_{t +}^{0}$ , $\overline{F}_{t}^{0}$ and $F_{t}$ are kept distinct in every statement below. The strong Markov theorem is stated for $(F_{t})$ and the deterministic Markov theorems are stated for both $(F_{t}^{0})$ and $(F_{t})$ ; the companion examples page carries a counterexample showing that $F_{0 +}^{0} \neq F_{0}^{0}$ in the canonical realization.

The raw filtration remains raw even though the ambient measure has been completed; no null set is removed from $Ω$ . Choice is used for the completion theorem and the countable witnesses above, and is also inherited from the ambient Brownian construction.

Source notes

Sousi, Definition 6.10 (printed p. 54), defines the natural filtration and its raw right limit; it does not supply the completion construction. Completion is supplied by the declared measure-space theorem, with the ambient-null-ideal convention and the symmetric-difference description proved above. These distinctions are needed when moving between completed events and raw representatives in the later Markov and zero-one arguments.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

The Brownian transition semigroup

Definition

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion Brownian motion. For $t > 0$ define the Brownian transition kernel $p_{t} (x, y) : = (2 π t)^{- 1 / 2} \exp ⁣ (- \frac{(y - x)^{2}}{2 t}), x, y \in R,$ and for every bounded Borel function $f : R \to R$ define $P_{t} f (x) : = \int_{R} f (y) p_{t} (x, y) d y (t > 0), P_{0} f : = f .$ The family $(P_{t})_{t \geq 0}$ is the Brownian transition semigroup, and each $P_{t}$ is a transition operator.

Two equivalent descriptions are part of the definition and are used below.

Expectation form. For a standard Brownian motion $B$ as in Brownian motion and $t \geq 0$ , $P_{t} f (x) = E [f (x + B_{t})] .$ Under AC, $N (0, t)$ is the law of $\sqrt{t} Z$ for a standard normal $Z$ , whose density $φ (z) = e^{- z^{2} / 2} / \sqrt{2 π}$ is fixed in Standard normal and normal laws; the density of $N (x, t)$ is the translate $y \mapsto p_{t} (x, y)$ , and the agreement of the two displayed expressions is proved as the first assertion of the semigroup lemma later on this page.
Basic regularity. For $t > 0$ the map $(x, y) \mapsto p_{t} (x, y)$ is continuous, hence Borel; consequently $P_{t} f$ is Borel for bounded Borel $f$ , $P_{t}$ is linear, and $∥ P_{t} f ∥_{\infty} \leq ∥ f ∥_{\infty}$ , with equality for $f \equiv 1$ once the kernel is known to be a probability density. At $t = 0$ the convention is $P_{0} f = f$ , so $P_{0}$ is the identity.

The cases $t = 0$ and $s = 0$ of every later identity are the identity operator and are recorded separately rather than derived from the $t > 0$ formula. No choice beyond the declared AC is made by the kernel: the integral is a Lebesgue integral of a fixed continuous density.

Source notes

Lawler, Section 2.6, and Durrett, Section 7.3, define the Brownian transition density and the operator $P_{t} f$ . The expectation form is the definition of $P_{t}$ in Lawler's Markov-viewpoint treatment; here it is stated as an equivalent description and proved in the following lemma, so that no step of the later arguments has to treat it as an extra hypothesis.

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

The Brownian kernels form a semigroup

Statement

Assume the Axiom of Choice, and let $p_{t}$ and $P_{t}$ be the Brownian transition kernel and operators of The Brownian transition semigroup.

Expectation form. If $B$ is a standard Brownian motion Brownian motion, then for every $t \geq 0$ , every $x \in R$ and every bounded Borel $f : R \to R$ , $P_{t} f (x) = E [f (x + B_{t})] .$
Semigroup identity. For all $s, t \geq 0$ , $P_{s} P_{t} = P_{s + t}$ as operators on bounded Borel functions.
Kernel identity. For all $s, t > 0$ and all $x, z \in R$ , $\int_{R} p_{s} (x, y) p_{t} (y, z) d y = p_{s + t} (x, z) .$ Conversely, the kernel identity for all $x, z$ implies the semigroup identity for all bounded Borel $f$ .
Probability kernels. $P_{t} 1 = 1$ for every $t \geq 0$ , so each $P_{t}$ is a probability kernel operator and $∥ P_{t} f ∥_{\infty} \leq ∥ f ∥_{\infty}$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , bounded Borel $f$ , and $s, t \geq 0$ .

[F1]

$B_{0} = 0$ almost surely, and for every finite list $0 = t_{0} < t_{1} < \dots < t_{n}$ the increments are independent with laws $N (0, t_{j} - t_{j - 1})$ . Brownian motion

[F2]

$N (0, 1)$ is the measure $γ$ with density $φ (y) = e^{- y^{2} / 2} / \sqrt{2 π}$ , and for $m \in R$ , $σ \geq 0$ the law $N (m, σ^{2})$ is the pushforward of $γ$ under $x \mapsto m + σ x$ ; $φ$ is positive with integral one. Standard normal and normal laws The standard normal density has total mass one

[F3]

For an affine increasing $C^{1}$ substitution with continuous outer integrand, oriented compact substitution holds; the nonnegative integrals on $R$ are the increasing limits of their compact restrictions. On each compact interval the continuous integrands are bounded and Riemann integrable, and their Riemann and Lebesgue integrals agree under countable choice (supplied by AC). A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ Monotone convergence for the integral

[F4]

A probability measure on $R$ is determined by its distribution function: two Borel probability measures with the same values on the intervals $(- \infty, y]$ coincide. This uses countable choice. Probability laws correspond to distribution functions

[F5]

Countable choice is the restriction of AC to families indexed by the natural numbers, so the AC assumption gives it directly. The Axiom of Countable Choice ( ${A C}_{ω}$ ) The Axiom of Choice

[F6]

A nonnegative measurable density defines a measure, and integration against that measure is integration of the product with the density. The indefinite integral of a nonnegative measurable function is a measure Integrating against a density agrees with integrating the product

[F7]

$\int_{R} e^{- x^{2}} d x = \sqrt{π}$ . The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$

[F8]

Tonelli applies to nonnegative product-measurable integrands. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Proof

technique · direct

1.1

Fix $m \in R$ and $σ > 0$ . Write $γ_{s} (y) : = (2 π s)^{- 1 / 2} \exp (- y^{2} / (2 s))$ for $s > 0$ . Since $N (m, σ^{2})$ is by [F2] the law of $m + σ Z$ with $Z \sim γ$ , its distribution function at $y$ is $P (m + σ Z \leq y) = P (Z \leq (y - m) / σ) = \int_{- \infty}^{(y - m) / σ} φ (u) d u$ ; for $L > \max (0, (m - y) / σ)$ , [F3] applied to the increasing affine map $u = (v - m) / σ$ on $[m - σ L, y]$ gives $\int_{m - σ L}^{y} γ_{σ^{2}} (v - m) d v = \int_{- L}^{(y - m) / σ} φ (u) d u$ , and letting $L = L_{0} + n \to \infty$ with $L_{0} = 1 + \max (0, (m - y) / σ)$ and $n \in N$ , [F3]'s monotone convergence identifies the distribution function of $m + σ Z$ with $\int_{- \infty}^{y} γ_{σ^{2}} (v - m) d v$ , that is with that of the measure with density $v \mapsto γ_{σ^{2}} (v - m)$ .

F2F3

2.1

The density measure in step 1.1 exists by [F6]. Its total mass is one: let $y \to \infty$ through positive integers in the established half-line identity and use [F3] and the mass-one assertion of [F2]. By [F4], with countable choice supplied by [F5], the density of step 1.1 represents $N (m, σ^{2})$ when $σ > 0$ , and by [F6] expectations against that law are integrals of the product with the density.

F2F3F4F5F6step 1.1

3.1

By [F1] with the one-term list $0 < t$ , the random variable $B_{t} - B_{0}$ has law $N (0, t)$ ; since $B_{0} = 0$ almost surely, $B_{t}$ has the same law $N (0, t)$ . By steps 1.1-2.1 with $m = 0$ and $σ = \sqrt{t}$ , the law of $B_{t}$ has density $γ_{t}$ , and the law of $x + B_{t}$ has density $v \mapsto γ_{t} (v - x) = p_{t} (x, v)$ ; hence for bounded Borel $f$ , $P_{t} f (x) = \int_{R} f (v) p_{t} (x, v) d v = E [f (x + B_{t})]$ , while for $t = 0$ both sides equal $f (x)$ by the convention $P_{0} f = f$ and $B_{0} = 0$ almost surely. This is assertion 1.

F1F6step 1.1step 2.1

4.1

Continuing the kernel analysis of step 3.1, fix $s, t > 0$ and $x, z \in R$ and put $A : = \frac{1}{2 s} + \frac{1}{2 t} = \frac{s + t}{2 s t}$ , $m : = \frac{t x + s z}{s + t}$ and $C : = \frac{(z - x)^{2}}{2 (s + t)}$ .

step 3.1algebra

5.1

Expanding squares gives $\frac{(y - x)^{2}}{2 s} + \frac{(z - y)^{2}}{2 t} = A (y - m)^{2} + C$ : $A$ is the coefficient of $y^{2}$ , $2 A m = x / s + z / t$ is the coefficient of $- y$ , and the constant coefficient identity is $A m^{2} + C = x^{2} / (2 s) + z^{2} / (2 t)$ ; consequently $p_{s} (x, y) p_{t} (y, z) = \frac{1}{2 π \sqrt{s t}} \exp (- A (y - m)^{2} - C)$ .

step 4.1algebra

6.1

For $L > 0$ , [F3] applied to the increasing affine map $v = \sqrt{A} (y - m)$ on $[m - L / \sqrt{A}, m + L / \sqrt{A}]$ converts the Gaussian integral [F7] into $\int_{m - L / \sqrt{A}}^{m + L / \sqrt{A}} e^{- A (y - m)^{2}} d y = A^{- 1 / 2} \int_{- L}^{L} e^{- v^{2}} d v$ ; letting $L \to \infty$ with [F3]'s monotone convergence and [F7] gives $\int_{R} e^{- A (y - m)^{2}} d y = \sqrt{π / A} = \sqrt{2 π s t / (s + t)}$ . Multiplying by the constant of step 5.1, $\int_{R} p_{s} (x, y) p_{t} (y, z) d y = \frac{1}{2 π \sqrt{s t}} \sqrt{\frac{2 π s t}{s + t}} e^{- C} = \frac{1}{\sqrt{2 π (s + t)}} e^{- (z - x)^{2} / (2 (s + t))} = p_{s + t} (x, z)$ , which is the kernel identity of assertion 3.

F3F7step 5.1algebra

7.1

Take bounded Borel $f \geq 0$ and $s, t > 0$ . For fixed $t > 0$ , Tonelli [F8] on the Borel Lebesgue measure spaces shows that $y \mapsto \int f (z) p_{t} (y, z) d z$ is Borel, since $(y, z) \mapsto f (z) p_{t} (y, z)$ is nonnegative and product-measurable. Its absolute value is at most $∥ f ∥_{\infty}$ by the density mass in step 2.1. Thus $P_{t} f$ is bounded Borel and $P_{s} (P_{t} f) (x) = \int_{R} (\int_{R} f (z) p_{t} (y, z) d z) p_{s} (x, y) d y$ ; the integrand is nonnegative and product-measurable, so [F8] rewrites the iterated integral as $\int_{R} f (z) (\int_{R} p_{s} (x, y) p_{t} (y, z) d y) d z = \int_{R} f (z) p_{s + t} (x, z) d z = P_{s + t} f (x)$ , using step 6.1 for the inner integral. Applying this to $f^{+}$ and $f^{-}$ and subtracting extends it to general bounded Borel $f$ , all four integrals being finite; if $s = 0$ or $t = 0$ both sides are $P_{t} f$ or $P_{s} f$ by the convention $P_{0} f = f$ .

F8step 2.1step 3.1step 6.1

8.1

Taking $f = 1$ in step 3.1 gives $P_{t} 1 (x) = \int_{R} p_{t} (x, y) d y = 1$ for $t > 0$ , and for $t = 0$ it is the convention, so every $P_{t}$ maps bounded Borel functions to bounded Borel functions with sup norm at most that of its argument; this is assertion 4. Assertion 1 is step 3.1, assertion 2 is step 7.1 and assertion 3 is steps 6.1 and 7.1. AC is used through the Brownian and normal-law interfaces of [F1]-[F2], and [F5] supplies the countable choice required both by the Riemann-to-Lebesgue conversion in [F3] (used in steps 1.1 and 6.1) and by the distribution-function uniqueness in [F4]. The substitution, Gaussian-integral and Tonelli interfaces make no additional choice beyond these declared uses.

F1F2F3F4F5step 1.1step 3.1step 6.1step 7.1∎

Source notes

The proof independently computes the Gaussian convolution by completing the square, then applies Tonelli. The cited stochastic-calculus sources provide the Brownian transition-kernel context; no exact completing-square computation in those sections is required as a premise.

LemmaStatement: AI-adaptedProof: AI-adaptedaudited 2026-09-22Open item page →

Conditioning a known state and independent noise

Statement

Assume the Axiom of Choice for the conditional-expectation interface. Let $(Ω, F, P)$ be a probability space, let $G \subseteq F$ be a sub-sigma-algebra, let $X$ be an $(S, Σ)$ -valued random element that is $G$ -measurable, and let $Y$ be a $(T, T)$ -valued random element whose sigma-algebra $σ (Y)$ is independent of $G$ . Let $μ$ be the law of $Y$ on $(T, T)$ . If $h : S \times T \to R$ is bounded and $Σ \otimes T$ -measurable, then $H (x) : = \int_{T} h (x, y) μ (d y)$ is $Σ$ -measurable and $E [h (X, Y) ∣ G] = H (X) almost surely .$

Facts & Assumptions

Given: AC, a probability space, a sub-sigma-algebra $G$ , a $G$ -measurable random element $X$ , a random element $Y$ with $σ (Y)$ independent of $G$ , and a bounded product-measurable $h$ .

[F1]

Measurability of $x \mapsto \int_{T} h (x, y) K (x, d y)$ for a finite kernel $K$ , in particular a probability kernel, is theorem-level; the constant map $K (s, A) : = μ (A)$ is a probability kernel because $s \mapsto μ (A)$ is constant. Measure kernel and probability kernel Measurability of integration against a kernel

[F2]

A conditional-expectation version is characterized by its $G$ -event integrals, and versions are unique almost surely. Conditional expectation given a sigma algebra Conditional expectation as an ae class Conditional expectation is unique almost surely

[F3]

Bounded $G$ -measurable factors come out of the conditional expectation, and conditional expectation is linear on integrable inputs. Taking out what is known Basic algebra and order properties of conditional expectation

[F4]

If a random variable has the independence rectangle identity against $G$ , its conditional expectation given $G$ is its mean; this applies to $1_{C} (Y)$ for $C \in T$ because $σ (Y)$ is independent of $G$ . Conditioning a known variable and an independent variable Independent sigma-algebras and independent events Independent random elements

[F5]

The product sigma-algebra is generated by the measurable rectangles, which form a pi-system containing the whole space; a lambda-system containing a pi-system contains the generated sigma-algebra. Dynkin's pi-lambda theorem The product sigma-algebra and its finite iterates

[F6]

Sections of product-measurable sets are measurable, and compositions of a measurable map with a measurable function are measurable. Every section of a product-measurable function is measurable Closure properties of measurable functions used by the integral Law or distribution of a random element

[F7]

Nonnegative measurable functions are increasing limits of nonnegative simple functions, and monotone convergence passes those limits through integrals. Every nonnegative measurable function is the increasing limit of simple measurable functions Monotone convergence for the integral

[F8]

AC supplies the conditional-expectation existence used in [F2]. The Axiom of Choice

Proof

technique · direct

1.1

Fix a measurable rectangle $A \times C$ with $A \in Σ$ and $C \in T$ . The indicator $1_{A \times C} (X, Y) = 1_{A} (X) 1_{C} (Y)$ has bounded $G$ -measurable factor $1_{A} (X)$ , so [F3] and then [F4] give $E [1_{A \times C} (X, Y) ∣ G] = 1_{A} (X) E [1_{C} (Y) ∣ G] = 1_{A} (X) P (Y \in C)$ almost surely; since $H_{A \times C} (x) = \int_{T} 1_{A \times C} (x, y) μ (d y) = μ (C) 1_{A} (x)$ , this is the asserted identity for rectangles.

F3F4given

2.1

Let $D$ be the class of $B \in Σ \otimes T$ with $E [1_{B} (X, Y) ∣ G] = H_{B} (X)$ almost surely, where $H_{B} (x) : = μ (B_{x})$ and $B_{x} = {y \in T : (x, y) \in B}$ . Each $B_{x}$ is measurable by [F6], the constant kernel $K (s, \cdot) : = μ$ is a probability kernel and [F1] makes $H_{B}$ $Σ$ -measurable, while $H_{B} (X)$ is then $G$ -measurable and bounded by [F6]. Step 1.1 shows that $D$ contains every measurable rectangle.

F1F6step 1.1

3.1

The class $D$ is a lambda-system on $S \times T$ . It contains $S \times T$ because $H_{S \times T} (x) = μ (T) = 1$ for every $x \in S$ , so both sides of its defining conditional-expectation identity are $1$ . If $B_{1} \subseteq B_{2}$ lie in $D$ , then for every $G$ -event $G_{0}$ , subtraction of their defining event-integral identities gives $\int_{G_{0}} 1_{B_{2} ∖ B_{1}} (X, Y) d P = \int_{G_{0}} (H_{B_{2}} - H_{B_{1}}) (X) d P$ , while sectionwise $H_{B_{2} ∖ B_{1}} = H_{B_{2}} - H_{B_{1}}$ ; hence $B_{2} ∖ B_{1} \in D$ by [F2]. If $B_{1} \subseteq B_{2} \subseteq \dots$ are in $D$ with union $B$ , then for each $x$ the numbers $H_{B_{n}} (x) = μ ((B_{n})_{x})$ increase to $μ (B_{x}) = H_{B} (x)$ , and [F7] applied to the finite measure $P$ restricted to $G_{0}$ gives $\int_{G_{0}} 1_{B_{n}} (X, Y) d P = \int_{G_{0}} H_{B_{n}} (X) d P \to \int_{G_{0}} H_{B} (X) d P$ for every $G$ -event $G_{0}$ ; hence $B \in D$ by [F2].

F2F7step 2.1

4.1

The measurable rectangles form a pi-system containing the product-space whole set $S \times T$ and generate $Σ \otimes T$ , so [F5] applied to the lambda-system $D$ gives $D = Σ \otimes T$ ; that is, $E [1_{B} (X, Y) ∣ G] = H_{B} (X)$ almost surely for every product-measurable $B$ .

F5step 2.1step 3.1

5.1

Let $h \geq 0$ be bounded. By [F7] there are nonnegative simple functions $s_{n} = \sum_{j} c_{j, n} 1_{B_{j, n}}$ with $s_{n} ↑ h$ pointwise, the sets $B_{j, n}$ being product measurable and the sums finite; step 4.1 and linearity of the integral give, for every $G$ -event $G_{0}$ , $\int_{G_{0}} s_{n} (X, Y) d P = \sum_{j} c_{j, n} \int_{G_{0}} 1_{B_{j, n}} (X, Y) d P = \sum_{j} c_{j, n} \int_{G_{0}} H_{B_{j, n}} (X) d P = \int_{G_{0}} H_{s_{n}} (X) d P$ , where $H_{s_{n}} (x) = \int_{T} s_{n} (x, y) μ (d y)$ .

F7step 4.1

6.1

For bounded nonnegative $h$ , monotone convergence [F7] applied to $s_{n} (X, Y) ↑ h (X, Y)$ and to $H_{s_{n}} (X) ↑ H_{h} (X)$ in step 5.1 gives $\int_{G_{0}} h (X, Y) d P = \int_{G_{0}} H_{h} (X) d P$ for every $G$ -event $G_{0}$ . Since $H_{h}$ is $Σ$ -measurable by [F1] and bounded by $∥ h ∥_{\infty}$ , [F2] identifies $H_{h} (X)$ with $E [h (X, Y) ∣ G]$ almost surely.

F1F2F7step 5.1

7.1

For a general bounded real $h$ , apply step 6.1 to the bounded nonnegative functions $h^{+}$ and $h^{-}$ and subtract the two almost-sure identities using linearity [F3]; since $H_{h} = H_{h^{+}} - H_{h^{-}}$ pointwise, $H_{h}$ is $Σ$ -measurable and $E [h (X, Y) ∣ G] = H_{h} (X)$ almost surely. This is the asserted identity, with $H = H_{h}$ as displayed in the statement.

F1F3step 6.1

8.1

The boundary cases behave as stated and need no separate treatment: $h \equiv 0$ gives $H \equiv 0$ and both sides vanish; if $T$ is a single point with its unique probability measure, then $H (x) = h (x, y_{0})$ and the statement reduces to $E [h (X, y_{0}) ∣ G] = h (X, y_{0})$ for the $G$ -measurable $X$ , which is [F3] with a deterministic factor; if $A = \emptyset$ or $C = \emptyset$ the rectangle identity of step 1.1 reads $0 = 0$ . The independence hypothesis is used exactly once, in step 1.1, and no other selection is made; AC is used only through [F8] in [F2].

F3F8givenstep 1.1∎

Source notes

Durrett's proof of Theorem 7.2.1 conditions on the known value of $B_{s}$ and on the independent increment; van der Vaart, Section 1.4, records the rectangle-to-product-sigma-algebra Dynkin extension in this generality. The proof above isolates that extension as a lemma because the Brownian Markov, future-path and planar arguments all consume it with different state spaces $S$ and $T$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Markov property of Brownian motion

Statement

Assume the Axiom of Choice, let $B$ be a standard Brownian motion with raw natural filtration $(F_{t}^{0})$ and usual augmentation $(F_{t})$ Natural and usual augmented Brownian filtrations, and let $p_{t}$ , $P_{t}$ be the Brownian transition kernel and operators The Brownian transition semigroup.

For all deterministic $s, t \geq 0$ and every bounded Borel $f : R \to R$ , $E [f (B_{s + t}) ∣ F_{s}^{0}] = P_{t} f (B_{s}) and E [f (B_{s + t}) ∣ F_{s}] = P_{t} f (B_{s})$ almost surely. Thus the Markov property holds both for the raw past and for the usual augmented past.

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , deterministic $s, t \geq 0$ , and a bounded Borel $f$ .

[F1]

Brownian increments along a finite strictly increasing list are mutually independent with laws $N (0, Δ t)$ , and $B_{0} = 0$ almost surely. Brownian motion

[F2]

Grouping a finite independent family of sigma-algebras by disjoint index sets gives independent generated sigma-algebras, and independent pi-systems containing the whole space generate independent sigma-algebras. Disjoint groups of an independent sigma-algebra family remain independent Independent pi-systems generate independent sigma-algebras Independent sigma-algebras and independent events

[F3]

For a $G$ -measurable random element $X$ and a random element $Y$ independent of $G$ with law $μ$ , the conditional expectation of $h (X, Y)$ is $H (X)$ with $H (x) = \int h (x, y) μ (d y)$ , for bounded product-measurable $h$ . Conditioning a known state and independent noise Independent random elements

[F4]

$P_{t} f (x) = E [f (x + B_{t})]$ for bounded Borel f and t>=0. For t>0 this equals $\int_{R} f (v) p_{t} (x, v) d v$ ; at t=0, $P_{0} f = f$ and there is no density p_0. The Brownian kernels form a semigroup The Brownian transition semigroup

[F5]

Conditional-expectation versions are characterized by their event integrals and are unique almost surely. Bounded pointwise-convergent random variables may be passed through event integrals by dominated convergence. Conditional expectation as an ae class Conditional expectation is unique almost surely Dominated convergence

[F6]

Every set in the completed raw sigma-algebra $\overline{F}_{u}^{0}$ differs from a set of $F_{u}^{0}$ by a subset of a $P$ -null event, and the two integrals of a bounded measurable function over such sets agree. Natural and usual augmented Brownian filtrations

[F7]

AC supplies the conditional-expectation interface of [F5]. The Axiom of Choice

Proof

technique · direct

1.1

First suppose t>0 and put Y=B_{s+t}-B_s. For any finite set of past times in [0,s], form their increasing union with {0,s,s+t}, delete repetitions, and apply [F1]. The past values differ almost surely from the cumulative sums of the increments up to s only by B_0=0; the final increment Y is independent of all those earlier increments by [F2]. For any Borel past cylinder its indicator agrees almost surely with the corresponding cylinder in those cumulative sums, so its joint probability with {Y in C} factors for every Borel C. This includes cylinders involving time zero and the case s=0, where all past cylinders have probability zero or one. Finite past cylinders generate the entire raw past; the pi-system criterion [F2] therefore makes sigma(Y) independent of that past. Finally Y has law N(0,t), also the law of B_t since B_0=0 almost surely.

F1F2given

2.1

Apply [F3] with $X = B_{s}$ , which is $F_{s}^{0}$ -measurable, with $Y$ as in step 1.1, and with $h (x, y) : = f (x + y)$ : the function $H (x) = \int_{R} f (x + y) μ (d y)$ with $μ = N (0, t)$ is Borel, and $E [f (B_{s + t}) ∣ F_{s}^{0}] = H (B_{s})$ almost surely. Since $μ$ is also the law of $B_{t}$ , step 1.1 and [F4] identify $H (x) = E [f (x + B_{t})] = P_{t} f (x)$ for every $x$ . This proves the raw-filtration assertion.

F3F4step 1.1

3.1

Suppose $t > 0$ and put $u_{n} = s + t / (n + 2)$ for integers n>=0, so $u_{n} ↓ s$ with $s < u_{n} < s + t$ . Put $h_{n} = s + t - u_{n} > 0$ . Applying step 2.1 at the time pair $(u_{n}, h_{n})$ and then completing the conditioning sigma-algebra as in [F6] gives $E [f (B_{s + t}) ∣ \overline{F}_{u_{n}}^{0}] = P_{h_{n}} f (B_{u_{n}})$ almost surely. Indeed, bounded event integrals are unchanged when an event is replaced by a raw event differing by a null set.

F5F6step 2.1

4.1

On the probability-one continuity event, $B_{u_{n}} \to B_{s}$ and $h_{n} \to t$ . The Gaussian densities $v \mapsto p_{h_{n}} (B_{u_{n}}, v)$ converge pointwise to $p_{t} (B_{s}, v)$ and, for all large $n$ , are bounded by an integrable envelope: choose a finite M bounding all the centers including B_s. Since $h_{n} \in [t / 2, t]$ , each density is at most $(π t)^{- 1 / 2} \exp (- ((∣ v ∣ - M)_{+})^{2} / (2 t))$ , which is integrable (bounded on [-M,M], with Gaussian tails). This bound may depend on the fixed path; that is sufficient for this pathwise integral limit. Dominated convergence therefore gives their $L^{1}$ convergence, and hence $P_{h_{n}} f (B_{u_{n}}) \to P_{t} f (B_{s})$ for bounded $f$ . If $A \in F_{s} = ⋂_{u > s} \overline{F}_{u}^{0}$ , then $A \in \overline{F}_{u_{n}}^{0}$ for every $n$ , so step 3.1 gives $\int_{A} f (B_{s + t}) d P = \int_{A} P_{h_{n}} f (B_{u_{n}}) d P$ . A second bounded dominated-convergence passage yields $\int_{A} f (B_{s + t}) d P = \int_{A} P_{t} f (B_{s}) d P$ . Since $P_{t} f (B_{s})$ is $F_{s}$ -measurable, [F5] proves the usual-filtration assertion.

F4F5F6step 3.1

5.1

For $t = 0$ the transition convention gives $P_{0} f = f$ and both identities read $E [f (B_{s}) ∣ G] = f (B_{s})$ , true because $B_{s}$ is $G$ -measurable for $G \in {F_{s}^{0}, F_{s}}$ . The proof also covers $s = 0$ when $t > 0$ , and then $P_{t} f (B_{0}) = P_{t} f (0)$ almost surely. If $f \equiv 0$ both sides vanish. AC is inherited from the Brownian and normal-law interfaces, the null-envelope witnesses in [F6], and in particular [F7] for conditional expectation; the independence and transition computations make no further choice.

F4F7givenstep 4.1∎

Source notes

Durrett, Theorem 7.2.1, proves the raw statement by conditioning on the known state and the independent increment; Sousi uses the same argument for the right-continuous filtration. The proof here separates the two filtrations explicitly and obtains the usual augmentation by conditioning at later raw times and passing those times down to the target time.

TheoremStatement: AI-adaptedProof: AI-adaptedaudited 2026-09-22Open item page →

Future-path Markov property

Statement

Assume the Axiom of Choice, let $B$ be a standard Brownian motion Brownian motion with raw natural filtration $(F_{t}^{0})$ and usual augmentation $(F_{t})$ Natural and usual augmented Brownian filtrations, and let $μ$ be Wiener measure on $C ([0, \infty), R)$ Wiener measure on continuous path space.

Write $u \mapsto u (t)$ for a point of the product space $R^{[0, \infty)}$ , whose product sigma-algebra is by definition generated by all finite coordinate cylinders (it exists by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal). Here Borel on the product means measurable for that sigma-algebra. The inclusion $j : C ([0, \infty), R) \to R^{[0, \infty)}$ is measurable because its coordinates are continuous. For a bounded Borel functional $Φ : R^{[0, \infty)} \to R$ put $Ψ_{Φ} (x) : = \int_{C ([0, \infty), R)} Φ (x + w) μ (d w), x \in R,$ where $x + w$ denotes the point $t \mapsto x + w (t)$ .

Increment process. For every $s \geq 0$ the random element $Z : = (B_{s + t} - B_{s})_{t \geq 0}$ is independent of $F_{s}$ and has the law of $B$ ; that is, for every finite cylinder event $Γ = {z : z (t_{1}) \in B_{1}, \dots, z (t_{k}) \in B_{k}}$ and every $A \in F_{s}$ , $P (A \cap {Z \in Γ}) = P (A) μ_{c y l} (Γ)$ , where $μ_{c y l} (Γ) = μ ({w : w (t_{i}) \in B_{i} \forall i})$ , and the analogous identity holds for every Borel set of the product sigma-algebra.
Conditional future law. For every $s \geq 0$ and every bounded Borel functional $Φ$ on $R^{[0, \infty)}$ , $E [Φ ((B_{s + t})_{t \geq 0}) ∣ F_{s}] = Ψ_{Φ} (B_{s}) almost surely,$ and the same identity holds with $F_{s}^{0}$ in place of $F_{s}$ . In particular $E [Φ ((B_{s + t})_{t \geq 0}) ∣ F_{s}]$ is a function of the single state $B_{s}$ .

The continuous-path formulation uses a common measurable probability-one event $A_{*}$ on which all paths of $B$ are continuous. Define $V^{s} (ω) (t) = B_{s + t} (ω)$ on $A_{*}$ and let $V^{s} (ω)$ be the zero path off $A_{*}$ . For every bounded Borel $ϕ : C ([0, \infty), R) \to R$ , the same conditional identity holds with $ϕ (V^{s})$ on the left and $\int ϕ (B_{s} + w) μ (d w)$ on the right, for either past sigma-algebra. This convention is independent almost surely of the choice of $A_{*}$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , $s \geq 0$ and a bounded Borel functional $Φ$ on $R^{[0, \infty)}$ .

[F1]

Brownian increments along finite strictly increasing lists are independent with laws $N (0, Δ t)$ , and $B_{0} = 0$ almost surely. Brownian motion

[F2]

Grouping finite independent families and the pi-system criterion for independent sigma-algebras. Disjoint groups of an independent sigma-algebra family remain independent Independent pi-systems generate independent sigma-algebras Independent sigma-algebras and independent events Independent random elements

[F3]

Conditioning a known state on independent noise: $E [h (X, Y) ∣ G] = H (X)$ for $H (x) = \int h (x, y) μ_{Y} (d y)$ . Conditioning a known state and independent noise

[F4]

Wiener measure is the law of a continuous Brownian motion, so its finite-dimensional marginals are the Brownian increment laws of [F1], and its Borel sigma-algebra is generated by the coordinates on a countable dense set. Wiener measure on continuous path space Borel sigma-algebra of continuous path space is generated by coordinates

[F5]

Conditional-expectation versions are characterized by event integrals and are unique almost surely; the tower identity holds for nested sigma-algebras. Conditional expectation as an ae class Conditional expectation is unique almost surely Tower property of conditional expectation

[F6]

A lambda-system containing a pi-system contains the generated sigma-algebra. By the convention in the statement, finite coordinate cylinders generate the product sigma-algebra; coordinate cylinders generate the Borel sigma-algebra of continuous path space. Dynkin's pi-lambda theorem Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal Borel sigma-algebra of continuous path space is generated by coordinates

[F7]

Simple approximation, monotone convergence and dominated convergence on the probability spaces used below; measurability of $x \mapsto \int Φ (x + w) μ (d w)$ by the constant-kernel integration theorem. Every nonnegative measurable function is the increasing limit of simple measurable functions Monotone convergence for the integral Dominated convergence Measurability of integration against a kernel Measure kernel and probability kernel

[F8]

Every set of the completed raw sigma-algebra $\overline{F}_{u}^{0}$ differs from a set of $F_{u}^{0}$ by a null set, and bounded integrals over the two sets agree; AC supplies the conditional-expectation interface. Natural and usual augmented Brownian filtrations The Axiom of Choice

Proof

technique · direct

1.1

Fix distinct times $0 \leq t_{1} < \dots < t_{k}$ and a bounded Borel $G : R^{k} \to R$ , and let $Φ$ be the cylinder functional $Φ (u) = G (u (t_{1}), \dots, u (t_{k}))$ . Then $Y : = (B_{s + t_{i}} - B_{s})_{i \leq k}$ is a random element of $R^{k}$ independent of $F_{s}^{0}$ with the law $ν$ of $(W_{t_{i}})_{i \leq k}$ under Wiener measure: independence follows as in the finite-cylinder argument by deleting repetitions, expressing $(B_{s}, B_{s + t_{1}}, \dots, B_{s + t_{k}})$ through the independent increments of [F1] and applying [F2]; for any finite past times $r_{j} \leq s$ , include the $r_{j}$ , $s$ and the $s + t_{i}$ in a common ordered grid. The increments before and after $s$ are independent, while $B_{0} = 0$ almost surely causes no change in event probabilities. This proves independence from every finite past cylinder, and [F2] extends it to their generated sigma-algebra $F_{s}^{0}$ . The law identity holds because both $Y$ and $(W_{t_{i}})$ are obtained from independent $N (0, t_{i} - t_{i - 1})$ increments (with $t_{0} = 0$ ) by the same cumulative-sum map, the Wiener marginal being [F4].

F1F2F4

2.1

With $Φ$ and $Y$ as in step 1.1 apply [F3] to $X = B_{s}$ , the sigma-algebra $F_{s}^{0}$ , the noise $Y$ , and $h (x, y) : = G (x + y_{1}, \dots, x + y_{k})$ : $H (x) = \int G (x + y_{1}, \dots, x + y_{k}) ν (d y)$ is Borel and $E [Φ ((B_{s + t})_{t \geq 0}) ∣ F_{s}^{0}] = H (B_{s})$ almost surely. By step 1.1's law identity, $H (x) = \int Φ (x + w) μ (d w) = Ψ_{Φ} (x)$ for every $x$ , because $Φ (x + w) = G (x + w (t_{1}), \dots, x + w (t_{k}))$ and the marginal of $(w (t_{i}))$ under $μ$ is $ν$ .

F3F4step 1.1

3.1

Let $D$ be the class of product-measurable sets $Γ \subseteq R^{[0, \infty)}$ such that $E [1_{Γ} ((B_{s + t})_{t \geq 0}) ∣ F_{s}^{0}] = μ_{Γ} (B_{s})$ almost surely, where $μ_{Γ} (x) : = μ (x + W \in Γ)$ . Each $μ_{Γ}$ is Borel by [F7] applied to the constant kernel $μ$ , since $(x, w) \mapsto x + w$ is product measurable. Finite cylinder sets lie in $D$ by step 2.1, and $D$ is a lambda-system: it contains the whole space because $μ_{R^{[0, \infty)}} \equiv 1$ ; it is closed under complements because $μ_{Γ^{c}} = 1 - μ_{Γ}$ and conditional expectations are additive on bounded inputs; and for disjoint sets in the class, countable additivity of the kernel and monotone convergence of the nonnegative finite sums give the event-integral identity for their union. Thus it is closed under disjoint countable unions, as required for a lambda-system.

F3F5F7step 2.1

4.1

The finite cylinder sets are a pi-system containing the whole space and generate the product sigma-algebra, so [F6] gives $D =$ all product-measurable sets: for every product-measurable $Γ$ , $E [1_{Γ} ((B_{s + t})_{t \geq 0}) ∣ F_{s}^{0}] = μ_{Γ} (B_{s})$ almost surely.

F6step 3.1

5.1

For bounded nonnegative $Φ$ , choose simple functionals $s_{n} ↑ Φ$ and use step 4.1 together with linearity of the integral to get $\int_{A} s_{n} ((B_{s + t})) d P = \int_{A} Ψ_{s_{n}} (B_{s}) d P$ for every $A \in F_{s}^{0}$ ; monotone convergence [F7] applied to both sides gives $\int_{A} Φ ((B_{s + t})) d P = \int_{A} Ψ_{Φ} (B_{s}) d P$ . Since $Ψ_{Φ} (B_{s})$ is bounded and $F_{s}^{0}$ -measurable, [F5] gives $E [Φ ((B_{s + t})) ∣ F_{s}^{0}] = Ψ_{Φ} (B_{s})$ almost surely, and splitting a bounded real $Φ$ into positive and negative parts extends this to all bounded Borel $Φ$ . This is the raw-filtration half of assertion 2.

F5F7step 4.1

6.1

We next prove the identity for the usual augmentation, first for a bounded continuous cylinder $Φ (u) = G (u (t_{1}), \dots, u (t_{k}))$ . Set $u_{n} = s + 1 / n$ . For $A \in F_{s} \subseteq \overline{F}_{u_{n}}^{0}$ , the raw identity of step 5.1 at time $u_{n}$ extends from $F_{u_{n}}^{0}$ to its completion by [F8], and gives $\int_{A} Φ ((B_{u_{n} + t})_{t \geq 0}) d P = \int_{A} Ψ_{Φ} (B_{u_{n}}) d P$ . Brownian continuity makes the left integrand converge almost surely to $Φ ((B_{s + t})_{t \geq 0})$ . Also $B_{u_{n}} \to B_{s}$ almost surely and $Ψ_{Φ}$ is continuous, because bounded convergence under Wiener measure applies to $Φ (x_{n} + w) \to Φ (x + w)$ for a continuous cylinder. Dominated convergence therefore yields $\int_{A} Φ ((B_{s + t})) d P = \int_{A} Ψ_{Φ} (B_{s}) d P$ .

F1F4F5F7F8step 5.1

7.1

Approximate each half-line coordinate-cylinder indicator by decreasing bounded continuous cylinder functions using $g_{m, c} (x) = \max (0, 1 - m \max (x - c, 0))$ and finite products of these functions; dominated convergence in step 6.1 gives the same event-integral identity for all half-line cylinders. The class of product-measurable sets for which that identity holds for every $A \in F_{s}$ is a lambda-system by the same event-integral and disjoint-union calculation as step 3.1; half-line cylinders form a generating pi-system, so it is the whole product sigma-algebra by [F6]. Increasing simple approximation and monotone convergence then extend the identity to every bounded nonnegative Borel $Φ$ , and positive-minus-negative decomposition to every bounded real $Φ$ . Since $Ψ_{Φ} (B_{s})$ is bounded and $F_{s}$ -measurable by [F7], [F5] identifies it as $E [Φ ((B_{s + t})) ∣ F_{s}]$ . This completes assertion 2 without reversing the tower property.

F5F6F7step 6.1

8.1

For assertion 1, fix a finite cylinder $Γ = {z : z (t_{1}) \in B_{1}, \dots, z (t_{k}) \in B_{k}}$ and let $Φ_{Γ} (u) : = 1_{Γ} ((u (t) - u (0))_{t \geq 0})$ , a bounded Borel functional. For every $x$ , $Φ_{Γ} (x + w) = 1_{Γ} ((w (t) - w (0))_{t \geq 0})$ is independent of $x$ , so $Ψ_{Φ_{Γ}} (x) = μ_{c y l} (Γ)$ is the constant $q_{Γ}$ ; step 7.1 therefore gives $P (A \cap {Z \in Γ}) = q_{Γ} P (A)$ for every $A \in F_{s}$ , and $q_{Γ} = P (Z \in Γ)$ . Hence $σ (Z)$ is independent of $F_{s}$ : the class of product-measurable $Γ$ satisfying $P (A \cap {Z \in Γ}) = P (A) P (Z \in Γ)$ for all $A \in F_{s}$ is a lambda-system containing the cylinder pi-system, so equals the product sigma-algebra by [F6]. Its finite-dimensional marginals are those of $μ$ by step 1.1's law identity, so the law of $Z$ is the pushforward $j_{*} μ$ on the product sigma-algebra; this is the assertion that the increment process is a Brownian motion independent of the past.

F6step 7.1

8.2

For the continuous-path formulation, every coordinate of $V^{s}$ is ambient-measurable and every value is in continuous path space, so [F4] makes $V^{s}$ a Borel random element. For a continuous-path cylinder functional its evaluation at $V^{s}$ agrees almost surely with the corresponding product cylinder evaluated on the original future, simultaneously in every time on $A_{*}$ . Thus steps 2.1 and 6.1 give its raw and augmented event-integral identities. The translation map $(x, w) \mapsto x + w$ into continuous path space is measurable because all its coordinates are measurable and [F4] gives the target sigma-algebra. Use the half-line approximation of step 7.1 for the augmented identity, then apply the lambda-system calculation of step 3.1 to cylinder sets in continuous path space, then [F6] and the simple approximation of step 5.1, to obtain both identities for every bounded Borel $ϕ$ . This argument asserts no adaptedness of the normalized coordinates: the right side is measurable with respect to the past because it is a Borel function of the original $B_{s}$ . Two choices of $A_{*}$ give identical paths on their probability-one intersection, proving the asserted independence of the convention.

F3F4F5F6F7step 2.1step 3.1step 5.1step 6.1step 7.1

9.1

The degenerate and endpoint cases are covered and consistent: if $k = 0$ or $Φ$ is constant, both sides equal that constant; if $t_{1} = 0$ then $Φ (u) = G (u (0), u (t_{2}), \dots)$ and the coordinate $B_{s + 0} - B_{s} = 0$ , so $Y$ has a degenerate first coordinate and $ν$ charges it correspondingly, while $μ$ 's marginal at time $0$ is the point mass at $w (0)$ — the identity of step 1.1 remains valid because both laws are the same law of a vector with a deterministic coordinate; for $s = 0$ one has $B_{0} = 0$ almost surely, $Ψ_{Φ} (B_{0}) = Ψ_{Φ} (0)$ , and the past sigma-algebras $F_{0}^{0} = σ (B_{0})$ and $F_{0} = ⋂_{u > 0} \overline{F}_{u}^{0}$ are handled by steps 5.1 and 7.1 unchanged. AC is inherited from the Brownian and Wiener interfaces and from [F8], including ambient completion and the conditional-expectation interface. The time sequence is explicit and no pathwise selection is made.

F8givenstep 1.1step 7.1∎

Source notes

Durrett, Section 7.2, and Sousi's argument preceding Theorem 6.13, state the future-path Markov property for the completed filtration. The proof above separates the finite-cylinder identity, the Dynkin extension over future-path events, and the passage from the raw past to the usual augmentation, so that Blumenthal's law and the strong Markov theorem can cite exactly the half they need.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

The Brownian germ sigma-algebra at zero

Definition

Assume the Axiom of Choice and let $B$ be a standard Brownian motion with raw natural filtration $(F_{t}^{0})$ and usual augmentation $(F_{t})$ Natural and usual augmented Brownian filtrations. The germ sigma-algebra at zero is $F_{0 +}^{0} : = ⋂_{t > 0} F_{t}^{0} .$ It records the events observable at arbitrarily small positive times in the uncompleted filtration.

Three elementary descriptions are part of the definition.

Countable intersection. Since $t \mapsto F_{t}^{0}$ is increasing, $F_{0 +}^{0} = ⋂_{q \in Q, q > 0} F_{q}^{0}$ . Indeed every positive rational is a positive real, giving one inclusion, while for a real $t > 0$ one may choose a rational $q \in (0, t)$ and use $F_{q}^{0} \subseteq F_{t}^{0}$ , so the countable intersection is contained in every $F_{t}^{0}$ .
Position relative to the usual augmentation. $F_{0 +}^{0} \subseteq F_{0}$ . For $u > 0$ one has $F_{u / 2}^{0} \subseteq F_{u}^{0} \subseteq \overline{F}_{u}^{0}$ , so $F_{0 +}^{0} \subseteq F_{u / 2}^{0} \subseteq \overline{F}_{u}^{0}$ for every $u > 0$ ; intersecting over $u > 0$ gives the claim. In particular every germ event is an event of the usual sigma-algebra at time zero.
No completion is included. The definition uses the raw sigma-algebras. The completed and augmented objects of Natural and usual augmented Brownian filtrations are not substituted for them, and the companion examples page records a counterexample showing that $F_{0 +}^{0} \neq F_{0}^{0}$ in the canonical realization.

AC is declared only because the ambient filtration definition inherits the Brownian construction's assumption; the intersection and its two descriptions make no selection.

Source notes

Sousi, Definition 6.10 and Theorem 6.13, and Durrett, Theorem 7.2.3, use the germ sigma-algebra at zero as the home of Blumenthal's zero-one law. The countable-intersection description is the form in which the increasing filtration is used, and the containment in the time-zero usual sigma-algebra is recorded because the zero-one law consumes it.

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Blumenthal's zero-one law

Statement

Assume the Axiom of Choice, let $B$ be a standard Brownian motion, and let $F_{0 +}^{0}$ be the germ sigma-algebra at zero The Brownian germ sigma-algebra at zero. Then every event $A \in F_{0 +}^{0}$ has $P (A) \in {0, 1}$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , and an event $A \in F_{0 +}^{0}$ .

[F1]

The germ sigma-algebra is the intersection of the raw pasts at positive times, and it is contained in the usual time-zero sigma-algebra $F_{0}$ . The Brownian germ sigma-algebra at zero Natural and usual augmented Brownian filtrations

[F2]

For every bounded Borel functional $Φ$ of the future path and every $s \geq 0$ , $E [Φ ((B_{s + t})_{t \geq 0}) ∣ F_{s}] = Ψ_{Φ} (B_{s})$ almost surely, where $Ψ_{Φ} (x) = \int Φ (x + w) μ (d w)$ , and likewise with the raw past in place of $F_{s}$ . Future-path Markov property

[F3]

Conditional-expectation versions are characterized by their event integrals, and $B_{0} = 0$ almost surely. Conditional expectation as an ae class Conditional expectation is unique almost surely Brownian motion

[F4]

AC supplies the conditional-expectation interface of [F3]. The Axiom of Choice

Proof

technique · direct

1.1

Since $A \in F_{0 +}^{0} = ⋂_{t > 0} F_{t}^{0}$ , one has $A \in F_{t}^{0}$ for every $t > 0$ , hence $A \in σ (⋃_{t > 0} F_{t}^{0}) = F_{\infty}^{0} = σ (B_{t} : t \geq 0)$ ; equivalently $A = {B \in A_{0}}$ for the corresponding product-measurable set $A_{0}$ of the path space, that is, $A$ is an event of the sigma-algebra generated by the whole future path $(B_{t})_{t \geq 0}$ . By [F1] we also have $A \in F_{0}$ .

F1given

1.2

Fix any $F_{0}$ -event $A^{'}$ and any product-measurable path set $Γ \subseteq R^{[0, \infty)}$ . Let $j : C ([0, \infty), R) \to R^{[0, \infty)}$ be the measurable inclusion from [F2]. By [F2] at $s = 0$ for the bounded Borel functional $1_{Γ}$ of the future path, $E [1_{Γ} ((B_{t})_{t \geq 0}) ∣ F_{0}] = Ψ_{1_{Γ}} (B_{0})$ almost surely; because $B_{0} = 0$ almost surely and $μ$ is Wiener measure on the continuous path space, $Ψ_{1_{Γ}} (0) = \int_{C} 1_{Γ} (j (w)) μ (d w) = μ (j^{- 1} (Γ))$ . Thus the conditional expectation is the constant $μ (j^{- 1} (Γ))$ . The event-integral characterization in [F3] therefore gives $P (A^{'} \cap {(B_{t})_{t \geq 0} \in Γ}) = P (A^{'}) μ (j^{- 1} (Γ))$ ; in particular, the product-space future path is independent of the completed time-zero sigma-algebra.

F2F3given

2.1

Apply step 1.2 to $A^{'} = A$ and to the path set $Γ : = A_{0}$ supplied by step 1.1, so that ${(B_{t})_{t \geq 0} \in Γ} = A$ . Taking $A^{'} = Ω$ in step 1.2 also gives $P (A) = μ (j^{- 1} (Γ))$ . Hence $P (A) = P (A \cap A) = P (A) μ (j^{- 1} (Γ)) = P (A)^{2}$ , so $P (A) \in {0, 1}$ .

step 1.1step 1.2

3.1

Step 2.1 classifies the number $P (A)$ , not the set $A$ . For example, a Brownian realization may contain a null exceptional outcome at which $B_{0} \neq 0$ ; then ${B_{0} \neq 0} \in F_{0}^{0} \subseteq F_{0 +}^{0}$ can be nonempty and proper while having probability zero. No right-continuity of the raw filtration at zero is used, only the containment of the germ in the usual time-zero sigma-algebra and the future-path independence at $s = 0$ . AC is used only through [F4] in the conditional-expectation characterization of step 1.2.

F3F4step 2.1∎

Source notes

Durrett, Theorem 7.2.3, and Sousi, Theorem 6.13, prove the zero-one law from the independence of the future increments from the germ. The proof above consumes the future-path theorem at $s = 0$ in the completed filtration, so completion causes no gap, and it avoids the reverse-martingale formulation.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-22Open item page →

Continuous-time stopping times and stopped sigma-algebras

Definition

Let $(Ω, F, P)$ be a probability space and let $(F_{t})_{t \geq 0}$ be a continuous-time filtration Continuous-time filtrations and all-pairs martingales.

A map $τ : Ω \to [0, \infty]$ (infinite values allowed) is a stopping time for $(F_{t})$ when ${τ \leq t} \in F_{t} for every t \geq 0.$
For a stopping time $τ$ its stopped sigma-algebra is $F_{τ} : = {A \in F : A \cap {τ \leq t} \in F_{t} for every t \geq 0} .$

The following facts are used below, and each is a direct set computation.

(a) $F_{τ}$ is a sigma-algebra: $\emptyset \cap {τ \leq t} = \emptyset \in F_{t}$ ; for $A \in F_{τ}$ one has $A^{c} \cap {τ \leq t} = {τ \leq t} ∖ (A \cap {τ \leq t}) \in F_{t}$ ; and for a sequence $A_{n} \in F_{τ}$ the union satisfies $(⋃_{n} A_{n}) \cap {τ \leq t} = ⋃_{n} (A_{n} \cap {τ \leq t}) \in F_{t}$ . All operations are literal, not modulo null sets, and no completeness of the filtration is assumed. (b) If $τ \equiv t_{0} < \infty$ is deterministic, then $A \cap {τ \leq t}$ is $A$ for $t \geq t_{0}$ and $\emptyset$ for $t < t_{0}$ , so $F_{τ} = ⋂_{t \geq t_{0}} F_{t} = F_{t_{0}}$ , where the last equality follows because the intersection includes its least member $F_{t_{0}}$ . If $τ \equiv \infty$ , then every test event ${τ \leq t}$ is empty and $F_{τ} = F$ . (c) Suppose the filtration is right-continuous. Then $τ$ is a stopping time if and only if ${τ < t} \in F_{t}$ for every $t > 0$ . Indeed ${τ < t} = ⋃_{n \geq 1} {τ \leq t - \frac{1}{n}}$ (with the terms for $t - \frac{1}{n} < 0$ read as the empty set, since $τ \geq 0$ ), which gives the forward implication; conversely ${τ \leq t} = ⋂_{n \geq 1} {τ < t + \frac{1}{n}},$ and $⋂_{n} F_{t + 1 / n} = ⋂_{s > t} F_{s} = F_{t}$ by right-continuity together with monotonicity of the filtration. (d) Under the same right-continuity assumption, for $A \in F$ one has $A \in F_{τ}$ if and only if $A \cap {τ < t} \in F_{t}$ for every $t > 0$ : the forward implication follows from $A \cap {τ < t} = ⋃_{n} (A \cap {τ \leq t - \frac{1}{n}})$ and the converse from $A \cap {τ \leq t} = ⋂_{n} (A \cap {τ < t + \frac{1}{n}})$ together with $⋂_{n} F_{t + 1 / n} = F_{t}$ .

Because the two versions of each test are interchangeable exactly when the filtration is right-continuous, the convention is recorded here once: the Brownian strong Markov theorem is stated for the usual augmentation, which is right-continuous, and the raw-filtration statements use the non-strict test directly. No choice principle is used by this definition.

Source notes

Durrett, Section 7.3, and Sousi, Section 6.4, define stopping times by ${τ \leq t} \in F_{t}$ and record the strict-test form for right-continuous filtrations. The stopped sigma-algebra convention is fixed here because the strong Markov theorem and the dyadic ceiling argument both consume the containment $F_{τ} \subseteq F_{τ_{n}}$ for $τ_{n} ↓ τ$ .

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Brownian closed-set hitting times are stopping times

Statement

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion all of whose paths are continuous, as in the canonical path-space realization of Wiener measure on continuous path space. Let $C \subseteq R$ be a closed set, with $dist (x, \emptyset) : = + \infty$ , and put $τ_{C} : = \inf {t \geq 0 : B_{t} \in C}, \inf \emptyset : = + \infty .$ Then $τ_{C}$ is a stopping time for the raw natural filtration $(F_{t}^{0})$ Natural and usual augmented Brownian filtrations; consequently it is a stopping time for the usual augmentation $(F_{t})$ and for every filtration containing $(F_{t}^{0})$ . The conclusion uses neither right-continuity of the filtration nor a separation assumption at time zero.

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ with everywhere continuous paths, and a closed set $C \subseteq R$ .

[F1]

A stopping time for a filtration is a map $τ$ with ${τ \leq t} \in F_{t}$ for all $t \geq 0$ ; larger filtrations keep the property. Continuous-time stopping times and stopped sigma-algebras Continuous-time filtrations and all-pairs martingales

[F2]

The raw natural filtration is $F_{t}^{0} = σ (B_{s} : 0 \leq s \leq t)$ , and the usual augmentation contains it. Natural and usual augmented Brownian filtrations

[F3]

The explicit hypothesis gives continuity of every path. For nonempty closed $C$ , put $d_{C} (x) = \inf_{z \in C} ∣ x - z ∣$ . This is a finite nonnegative real. Taking infima in $∣ x - z ∣ \leq ∣ x - y ∣ + ∣ y - z ∣$ gives $d_{C} (x) \leq ∣ x - y ∣ + d_{C} (y)$ ; swapping $x, y$ proves $∣ d_{C} (x) - d_{C} (y) ∣ \leq ∣ x - y ∣$ , hence continuity. For $x \in C$ , $d_{C} (x) = 0$ . For $x \notin C$ , the open complement contains $(x - r, x + r)$ for some $r > 0$ , so $d_{C} (x) \geq r > 0$ . Thus $d_{C} (x) = 0$ exactly on $C$ . Open subset of $R$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen

[F4]

Every bounded real sequence has a convergent subsequence; a limit of points of $[0, t]$ lies in $[0, t]$ , since a limit strictly outside would eventually force the points outside. Rationals lie strictly between any two distinct reals. Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence The rationals embed densely in the reals

[F5]

The canonical coordinate process under Wiener measure is a standard Brownian motion with continuous paths. Wiener measure on continuous path space Existence of continuous Brownian motion

[F6]

AC is declared because the ambient Brownian construction assumes it. The Axiom of Choice

Proof

technique · direct

1.1

If $C = \emptyset$ , then $τ_{C} = \infty$ and all its finite-time test events are empty; this case is settled. Now assume $C \neq \emptyset$ . Fix $t \geq 0$ and put $G_{t} : = ⋂_{n \geq 1} ⋃_{q \in Q \cap [0, t]} {dist (B_{q}, C) < 1 / n}$ . For a point of $G_{t}$ , use the declared AC to select $q_{n} \in Q \cap [0, t]$ with $dist (B_{q_{n}}, C) < 1 / n$ for every $n \geq 1$ . Apply [F4] to the bounded sequence $(q_{j + 1})_{j \in N}$ , obtaining a subsequence $q_{n_{k}}$ converges to some $q^{*} \in [0, t]$ ; continuity of the path and of the distance give $0 \leq dist (B_{q^{*}}, C) \leq {lim inf}_{k} (dist (B_{q^{*}}, C) - dist (B_{q_{n_{k}}}, C)) + {lim sup}_{k} 1 / n_{k} = 0$ , hence $dist (B_{q^{*}}, C) = 0$ and $B_{q^{*}} \in C$ by [F3], so $τ_{C} \leq q^{*} \leq t$ . Thus $G_{t} \subseteq {τ_{C} \leq t}$ .

F3F4F6given

2.1

Conversely suppose $τ_{C} \leq t$ . The hitting set is nonempty and bounded below. By its infimum property choose, using AC, a hit $s_{n} \in [τ_{C}, τ_{C} + 1 / n)$ for each $n \geq 1$ . Then $s_{n} \to τ_{C}$ , so continuity and [F3] imply $d_{C} (B_{τ_{C}}) = \lim_{n} d_{C} (B_{s_{n}}) = 0$ ; hence $B_{τ_{C}} \in C$ . Given $n \geq 1$ , continuity at $s = τ_{C}$ and rational density supply $q \in Q \cap [0, t]$ with $∣ B_{q} - B_{s} ∣ < 1 / n$ : use an interior rational sufficiently near $s$ when $s > 0$ , and $q = 0$ when $s = 0$ . Thus $d_{C} (B_{q}) \leq ∣ B_{q} - B_{s} ∣ < 1 / n$ . This proves ${τ_{C} \leq t} \subseteq G_{t}$ . Together with step 1.1 it yields equality for every $t \geq 0$ . At $t = 0$ , $G_{0} = ⋂_{n \geq 1} {d_{C} (B_{0}) < 1 / n} = {B_{0} \in C}$ .

F3F4F6step 1.1

3.1

Each set ${dist (B_{q}, C) < 1 / n}$ is in $F_{q}^{0} \subseteq F_{t}^{0}$ for $q \leq t$ , because $dist (\cdot, C)$ is continuous hence Borel and $B_{q}$ is $F_{q}^{0}$ -measurable; the union over the countable set $Q \cap [0, t]$ and the intersection over $n$ are therefore in $F_{t}^{0}$ . By step 2.1, ${τ_{C} \leq t} \in F_{t}^{0}$ for every $t \geq 0$ , so $τ_{C}$ is a stopping time for $(F_{t}^{0})$ by [F1], and hence for $(F_{t})$ by [F2].

F1F2F3step 2.1

4.1

The result concerns the given everywhere-continuous process; [F5] supplies the canonical realization as an example. No transfer to another raw natural filtration by null modification is used. The usual-augmentation conclusion follows solely by inclusion in step 3.1. For $C = R$ , $τ_{C} = 0$ and $G_{t} = Ω$ ; the empty-set case was settled in step 1.1. AC supplies the countable selections of approximating rational times and hit times made in steps 1.1 and 2.1, as well as the ambient construction [F6].

F5F6step 1.1step 2.1step 3.1∎

Source notes

The proof supplies the exact rational-distance formula for a closed subset of the real line and an everywhere-continuous process. The listed probability texts provide background on hitting times; no extension from an arbitrary almost-surely continuous version by terminal-null completion is claimed.

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Strong Markov property of Brownian motion

Statement

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion Brownian motion with usual augmentation $(F_{t})$ Natural and usual augmented Brownian filtrations, and let $τ$ be a stopping time for $(F_{t})$ with $τ < \infty$ almost surely Continuous-time stopping times and stopped sigma-algebras. Let $μ$ be Wiener measure Wiener measure on continuous path space and, for a bounded Borel functional $Φ$ on the product space $R^{[0, \infty)}$ , put $Ψ_{Φ} (x) : = \int_{C ([0, \infty), R)} Φ (x + w) μ (d w), x \in R .$ Here the product sigma-algebra is, by definition, the sigma-algebra generated by all finite coordinate cylinders. In this theorem "Borel functional" means measurable for this product sigma-algebra, not the possibly larger Borel sigma-algebra of the uncountable product topology.

Use the following measurable-version convention at random times. Put $τ_{n} = 2^{- n} ⌈ 2^{n} τ ⌉$ on ${τ < \infty}$ and $τ_{n} = \infty$ otherwise. For each $u \geq 0$ , let $V_{u}$ be the finite limit of $B_{τ_{n} + u}$ if that limit exists and $τ < \infty$ , and zero otherwise; each approximating variable is set to zero when $τ_{n} = \infty$ . Write $B_{τ} = V_{0}$ and $(B_{τ + u})_{u \geq 0} = V$ in the assertions below. On one measurable probability-one event these agree with the literal path values simultaneously for all $u$ , by path continuity. On an everywhere-continuous realization this convention changes only the event ${τ = \infty}$ . Under the usual augmentation the exceptional ambient null event belongs to $F_{0}$ , so this normalization preserves adaptedness as well as all almost-sure identities.

Then:

$B_{τ}$ is $F_{τ}$ -measurable, and the increment process $Z : = (B_{τ + t} - B_{τ})_{t \geq 0}$ is independent of $F_{τ}$ ; its finite-dimensional marginals are those of Wiener measure.
For every bounded Borel functional $Φ$ on $R^{[0, \infty)}$ , $E [Φ ((B_{τ + t})_{t \geq 0}) ∣ F_{τ}] = Ψ_{Φ} (B_{τ}) almost surely .$ Equivalently, the conditional law of the shifted future path given $F_{τ}$ is Wiener measure translated by $B_{τ}$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , a stopping time $τ$ for $(F_{t})$ with $τ < \infty$ almost surely, and a bounded Borel functional $Φ$ .

[F1]

Stopping time, stopped sigma-algebra, the strict-test description for right-continuous filtrations, and the containment $F_{τ} \subseteq F_{τ_{n}}$ for a decreasing family $τ_{n} ↓ τ$ . Continuous-time stopping times and stopped sigma-algebras Natural and usual augmented Brownian filtrations

[F2]

The usual augmentation is right-continuous and contains the raw filtration $F_{t}^{0} = σ (B_{s} : s \leq t)$ and every subset of every ambient null event. Natural and usual augmented Brownian filtrations

[F3]

For every $s \geq 0$ and every bounded Borel functional $Φ$ , $E [Φ ((B_{s + t})_{t \geq 0}) ∣ F_{s}] = Ψ_{Φ} (B_{s})$ almost surely, and the same holds with $F_{s}^{0}$ in place of $F_{s}$ . Future-path Markov property

[F4]

Brownian paths are continuous on a probability-one event; by the product-topology convention in the statement, coordinatewise convergence is convergence in the product topology, and continuous functions preserve it. Brownian motion

[F5]

Conditional-expectation versions are characterized by their event integrals and are unique almost surely; monotone and dominated convergence pass limits through integrals; nonnegative Borel functions are increasing limits of nonnegative simple functions; bounded real functions are handled by positive and negative parts. Conditional expectation as an ae class Conditional expectation is unique almost surely Monotone convergence for the integral Dominated convergence Every nonnegative measurable function is the increasing limit of simple measurable functions

[F6]

By the product-sigma convention in the statement, the half-line coordinate cylinders ${z : z (t_{1}) \leq c_{1}, \dots, z (t_{k}) \leq c_{k}}$ together with the whole space (the empty cylinder) form a pi-system that generates the product sigma-algebra. A lambda-system containing a pi-system contains the generated sigma-algebra. Dynkin's pi-lambda theorem

[F7]

The shift map $(x, w) \mapsto x + w$ from $R \times C ([0, \infty), R)$ to the product measurable space is measurable: each coordinate is the continuous map $(x, w) \mapsto x + w (u)$ . Hence $x \mapsto Ψ_{Φ} (x)$ is Borel for bounded Borel $Φ$ , by the integration theorem for the constant probability kernel $μ$ ; Wiener measure is the law of a continuous Brownian motion. Measurability of integration against a kernel Measure kernel and probability kernel Wiener measure on continuous path space Borel sigma-algebra of continuous path space is generated by coordinates

[F9]

Finite pointwise limits and their existence sets are measurable; assigning zero where a finite limit fails to exist preserves measurability. Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable

[F8]

AC supplies the conditional-expectation interface of [F5]. The Axiom of Choice

Proof

technique · direct

1.1

For $n \geq 1$ define the dyadic ceiling $τ_{n} : = 2^{- n} ⌈ 2^{n} τ ⌉$ , with $τ_{n} : = \infty$ when $τ = \infty$ . Then $τ \leq τ_{n} \leq τ + 2^{- n}$ wherever $τ < \infty$ , so $τ_{n} ↓ τ$ almost surely, each $τ_{n}$ takes values in the countable set $2^{- n} Z_{\geq 0} \cup {\infty}$ , and each $τ_{n}$ is a stopping time for $(F_{t})$ : for $t \geq 0$ , ${τ_{n} \leq t} = {τ \leq 2^{- n} ⌊ 2^{n} t ⌋} \in F_{2^{- n} ⌊ 2^{n} t ⌋} \subseteq F_{t}$ . Moreover $F_{τ} \subseteq F_{τ_{n}}$ because for $A \in F_{τ}$ one has $A \cap {τ_{n} \leq t} = A \cap {τ \leq t} \cap {τ_{n} \leq t} \in F_{t}$ .

F1F2given

2.1

For a countably valued stopping time $ρ$ , the variable $B_{ρ}$ set to zero at infinity is $F_{ρ}$ -measurable: its Borel inverse image, intersected with ${ρ \leq t}$ , is $⋃_{r \leq t} ({ρ = r} \cap {B_{r} \in D}) \in F_{t}$ . For the dyadic ceilings, $⋂_{n} F_{τ_{n}} = F_{τ}$ . Indeed, if $A$ belongs to the intersection, then $A \cap {τ < t} = ⋃_{n} (A \cap {τ_{n} < t}) \in F_{t}$ ; the right-continuous strict-test criterion [F1] proves membership in $F_{τ}$ . Each tail $(B_{τ_{n}})_{n \geq m}$ is measurable for $F_{τ_{m}}$ by the inclusion in step 1.1. The normalized finite limit is therefore measurable for every $F_{τ_{m}}$ by [F9], hence for $F_{τ}$ . The event ${τ < \infty}$ belongs to each of these stopped sigma-algebras, so the stated zero convention preserves this conclusion. For $u \geq 0$ , $B_{τ_{n} + u}$ is an ambient-measurable countable sum over the values of $τ_{n}$ , and its normalized limit $V_{u}$ is ambient measurable by [F9]. Thus $V$ and $Z = (V_{u} - V_{0})_{u}$ are random elements of the product measurable space. On the common event of path continuity and finite $τ$ , these limits equal $B_{τ + u}$ simultaneously for every $u$ .

F1F2F4F9step 1.1

3.1

Let $ρ$ be a stopping time taking values in a countable set $D \subseteq [0, \infty)$ with $ρ < \infty$ almost surely; by the argument of step 2.1 with $ρ$ in place of $τ$ the values $B_{ρ}$ are $F_{ρ}$ -measurable and $Ψ_{Φ} (B_{ρ})$ is bounded and $F_{ρ}$ -measurable for every bounded Borel functional $Φ$ . For every $A \in F_{ρ}$ , $\int_{A} Φ ((B_{ρ + u})_{u \geq 0}) d P = \int_{A} Ψ_{Φ} (B_{ρ}) d P$ : for each $r \in D$ the event $A_{r} : = A \cap {ρ = r}$ lies in $F_{r}$ , and [F3] at time $r$ gives $\int_{A_{r}} Φ ((B_{r + u})_{u \geq 0}) d P = \int_{A_{r}} Ψ_{Φ} (B_{r}) d P$ ; on $A_{r}$ one has $(B_{ρ + u}) = (B_{r + u})$ and $B_{ρ} = B_{r}$ , and summing over the countable set $D$ gives the identity. By [F5]'s uniqueness, $E [Φ ((B_{ρ + u})) ∣ F_{ρ}] = Ψ_{Φ} (B_{ρ})$ almost surely.

F3F5F7givenstep 2.1

4.1

First let $Φ (z) = g (z (u_{1}), \dots, z (u_{k}))$ , where $g$ is a bounded continuous function on $R^{k}$ . These cylinder functionals are product-measurable. On the common continuity event, the coordinates $B_{τ_{n} + u_{i}}$ tend to $V_{u_{i}}$ , so $Φ ((B_{τ_{n} + u})) \to Φ (V)$ . Further, $Ψ_{Φ}$ is continuous: if $x_{n} \to x$ , its integrand $g (x_{n} + w (u_{1}), \dots, x_{n} + w (u_{k}))$ converges pointwise and is bounded uniformly, so dominated convergence applies. For $A \in F_{τ} \subseteq F_{τ_{n}}$ , step 3.1 at $ρ = τ_{n}$ and dominated convergence therefore give $\int_{A} Φ (V) d P = \int_{A} Ψ_{Φ} (V_{0}) d P$ . Null exceptional events contribute zero to these integrals; no assertion that they belong to the past filtration is used.

F4F5step 1.1step 2.1step 3.1

5.1

For $c \in R$ and $k \geq 1$ let $g_{k, c} (y) : = \max {0, \min {1, k (c - y) + 1}}$ ; then $g_{k, c}$ is continuous and bounded with $g_{k, c} ↓ 1_{(- \infty, c]}$ pointwise as $k \to \infty$ . Consequently, for a half-line cylinder $Γ = {z : z (t_{i}) \leq c_{i}, i \leq m}$ the functions $G_{k} (z) : = \prod_{i \leq m} g_{k, c_{i}} (z (t_{i}))$ are bounded, continuous on the product space, and decrease pointwise to $1_{Γ}$ . Applying step 4.1 to $G_{k}$ and passing to the limit with dominated convergence [F5] on both sides, using $G_{k} ((B_{τ + u})) ↓ 1_{Γ} ((B_{τ + u}))$ and $Ψ_{G_{k}} (B_{τ}) \to Ψ_{1_{Γ}} (B_{τ})$ pointwise, gives $\int_{A} 1_{Γ} ((B_{τ + u})) d P = \int_{A} Ψ_{1_{Γ}} (B_{τ}) d P$ for every $A \in F_{τ}$ .

F5step 4.1

6.1

Let $D$ be the class of product-measurable sets $Γ$ for which $\int_{A} 1_{Γ} ((B_{τ + u})) d P = \int_{A} Ψ_{1_{Γ}} (B_{τ}) d P$ for every $A \in F_{τ}$ . Then $D$ is a lambda-system: it contains the whole product space because $Ψ_{1} = Ψ_{Φ}$ for $Φ \equiv 1$ is the constant $1$ ; it is closed under complements by subtracting the two finite identities; and it is closed under countable disjoint unions: first add the identities for the first $m$ disjoint sets, then use [F5] to pass to their union by monotone convergence on both sides. By step 5.1 it contains every half-line cylinder, which together with the empty cylinder form a generating pi-system, so [F6] gives $D$ equal to the whole product sigma-algebra.

F5F6step 5.1

7.1

For a bounded nonnegative Borel $Φ$ with simple functionals $s_{m} ↑ Φ$ , step 6.1 and linearity of the integral give $\int_{A} s_{m} ((B_{τ + u})) d P = \int_{A} Ψ_{s_{m}} (B_{τ}) d P$ for every $A \in F_{τ}$ ; monotone convergence [F5] on both sides, using $Ψ_{s_{m}} (x) ↑ Ψ_{Φ} (x)$ pointwise and the Borel measurability of $Ψ_{Φ}$ from [F7], gives $\int_{A} Φ ((B_{τ + u})) d P = \int_{A} Ψ_{Φ} (B_{τ}) d P$ . Since $Ψ_{Φ} (B_{τ})$ is bounded and $F_{τ}$ -measurable by step 2.1 and [F7], [F5]'s uniqueness identifies it with $E [Φ ((B_{τ + u})) ∣ F_{τ}]$ ; splitting a bounded real $Φ$ into positive and negative parts extends the identity to all bounded Borel $Φ$ . This is assertion 2.

F5F7step 2.1step 6.1

8.1

For assertion 1, fix a cylinder $Γ_{0}$ and put $Φ_{0} (u) : = 1_{Γ_{0}} ((u (t_{i}) - u (0))_{i \leq k})$ with $Γ_{0}$ a Borel subset of the finite coordinate space; the coordinate $u (0)$ is the translation offset. For every $x$ one has $Φ_{0} (x + w) = 1_{Γ_{0}} ((w (t_{i}) - w (0))_{i \leq k})$ , independent of $x$ , so $Ψ_{Φ_{0}} (x) = μ_{c y l} (Γ_{0}) : = μ ({w : (w (t_{i}) - w (0))_{i \leq k} \in Γ_{0}})$ is a constant; step 7.1 gives $P (A \cap {Z \in Γ_{0}}) = μ_{c y l} (Γ_{0}) P (A)$ for every $A \in F_{τ}$ , and taking $A = Ω$ shows that the finite-dimensional marginals of $Z$ are those of Wiener measure. The class of product-measurable sets satisfying $P (A \cap {Z \in Γ}) = P (A) P (Z \in Γ)$ for all $A \in F_{τ}$ is a lambda-system containing the cylinder pi-system, hence by [F6] equals the product sigma-algebra, so $Z$ is independent of $F_{τ}$ and its law on the product sigma-algebra has the finite-dimensional marginals of Wiener measure.

F6step 2.1step 7.1

9.1

The degenerate cases are consistent with the proof: if $τ \equiv t$ is deterministic then $F_{τ} = F_{t}$ by [F1], and step 3.1 reduces to the deterministic future-path theorem [F3]; the null set ${τ = \infty}$ is handled by the convention $B_{τ} = 0$ there and all identities are asserted almost surely; if $Φ$ is constant, then $Ψ_{Φ}$ is that constant and both sides agree; a coordinate at time $0$ is the deterministic offset $B_{τ}$ and was covered in step 8.1; and the case $Φ \equiv 0$ gives $0 = 0$ . AC is declared for the Brownian and conditional-expectation interfaces [F8]; no additional selections are made.

F1F3F8givenstep 8.1∎

Source notes

Durrett, Theorem 7.3.9, approximates the stopping time by dyadic ceilings and passes to the limit through continuity; Sousi, Theorem 6.17, states the result for the right-continuous filtration. The proof above derives the general stopping-time identity from the deterministic future-path theorem by that approximation, extends it from continuous to half-line cylinders by monotone limits, and closes the product sigma-algebra with Dynkin's pi-lambda theorem; no regular-conditional-distribution theory is assumed.

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Brownian reflection principle

Statement

Assume the Axiom of Choice, let $B$ be a standard Brownian motion Brownian motion. All path operations below use a fixed everywhere-continuous, zero-start representative: choose a measurable probability-one event on which the paths are continuous and $B_{0} = 0$ , and replace $B$ by the zero path off this event, as in Wiener measure on continuous path space. Retain the notation $B$ for this representative and use its own raw natural filtration and usual augmentation. This preserves every finite-dimensional law; all maxima and hitting times below refer to this representative, without transferring stopping times between the raw filtrations of different versions. Let $a > 0$ and $T > 0$ , and put $M_{T} : = \sup_{0 \leq s \leq T} B_{s}, τ_{a} : = \inf {s \geq 0 : B_{s} = a} .$ Then:

The path reflected at $τ_{a}$ has Wiener law: for each $T > 0$ the process $(B_{u})_{0 \leq u \leq T}$ and its reflection ${\tilde{B}}_{u} : = B_{u}$ for $u \leq τ_{a} \land T$ , ${\tilde{B}}_{u} : = 2 a - B_{u}$ for $τ_{a} \land T < u \leq T$ , have the same law on the cylinder sigma-algebra $σ (π_{u} : u \in [0, T])$ of $R^{[0, T]}$ .
For every $b \leq a$ , $P (M_{T} \geq a, B_{T} \leq b) = P (B_{T} \geq 2 a - b),$ and in particular $P (M_{T} \geq a) = 2 P (B_{T} \geq a)$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ with every path continuous and $B_{0} = 0$ everywhere, real $a > 0$ , $T > 0$ and $b \leq a$ .

[F1]

$τ_{a}$ is a stopping time for the raw natural filtration and hence for the usual augmentation, which is right-continuous; and $M_{T} \geq a$ if and only if $τ_{a} \leq T$ . The combination $S : = τ_{a} \land T$ is a stopping time bounded by $T$ . Brownian closed-set hitting times are stopping times Continuous-time stopping times and stopped sigma-algebras Natural and usual augmented Brownian filtrations

[F2]

Strong Markov: for an a.s. finite stopping time $τ$ of the usual augmentation, the shifted increment process $(B_{τ + t} - B_{τ})_{t \geq 0}$ is independent of $F_{τ}$ with the finite-dimensional marginals of Wiener measure; moreover $E [Φ ((B_{τ + t})) ∣ F_{τ}] = Ψ_{Φ} (B_{τ})$ for bounded Borel path functionals $Φ$ . Strong Markov property of Brownian motion

[F3]

Wiener measure is the unique Borel probability on continuous path space whose coordinates are centered Gaussian with covariance $\min (s, t)$ ; hence the map $w \mapsto - w$ preserves Wiener measure, because it preserves every finite-dimensional centered Gaussian law with that covariance. Uniqueness of Wiener measure Wiener measure on continuous path space

[F4]

For each $t > 0$ the law of $B_{t}$ has the strictly positive density $p_{t} (0, \cdot)$ , so $P (B_{t} = c) = 0$ for every $c$ . The Brownian kernels form a semigroup

[F5]

By definition, the cylinder sigma-algebra $σ (π_{u} : u \in [0, T])$ is generated by finite-coordinate cylinder sets, which form a pi-system; a lambda-system containing that pi-system contains the generated sigma-algebra, and conditional laws agree almost surely when their defining conditional expectations agree. Dynkin's pi-lambda theorem Conditional expectation as an ae class

[F7]

If $R$ is $F_{S}$ -measurable and $W$ is an independent random element with law $μ$ , then for bounded product-measurable $h$ , $E [h (R, W) ∣ F_{S}] = \int h (R, w) μ (d w)$ . The Borel sigma-algebra of continuous path space is generated by coordinates; thus coordinate measurability and cylinder independence imply path-space measurability and independence for an everywhere-continuous random path. Conditioning a known state and independent noise Borel sigma-algebra of continuous path space is generated by coordinates

[F6]

AC supplies the conditional-expectation and strong-Markov interfaces. The Axiom of Choice Strong Markov property of Brownian motion

Proof

technique · direct

1.1

By [F1] the time $τ_{a}$ is a stopping time of the usual augmentation and $S = τ_{a} \land T$ is a stopping time bounded by the deterministic time $T$ , hence a.s. finite; on the event ${τ_{a} \leq T}$ one has $S = τ_{a}$ and $B_{S} = a$ by continuity, while on ${τ_{a} > T}$ one has $S = T$ . Also $M_{T} \geq a ⟺ τ_{a} \leq T$ by continuity of the path.

F1given

1.2

The law of any finite-dimensional marginal of $B$ is unchanged by the substitution $B \mapsto - B$ , and the cylinder pi-system determines a law on $R^{[0, T]}$ ; hence the law of $(B_{u})_{u \leq T}$ is invariant under negation, and the same holds for every process whose finite-dimensional marginals are those of Brownian motion, by [F5] and [F3].

F3F5

2.1

Define ${\tilde{B}}_{u} : = B_{u}$ for $u \leq S$ and ${\tilde{B}}_{u} : = 2 a - B_{u}$ for $S < u \leq T$ . Every coordinate is measurable by the stopping-time tests. The path is continuous: if $S < T$ , then $B_{S} = a$ , and if $S = T$ it is unchanged. For finitely many $u_{i} \in [0, T]$ , put $R = (S, B_{u_{1} \land S}, \dots, B_{u_{k} \land S})$ . This is $F_{S}$ -measurable: $S$ is measurable there by its test events, and [F2]'s stopped-value measurability at $u_{i} \land S$ , followed by the stopped-sigma-algebra inclusion for $u_{i} \land S \leq S$ , gives the other coordinates. Since paths are everywhere continuous and these times finite, the version convention in [F2] gives the literal values. The path $W (r) = B_{S + r} - B_{S}$ is everywhere continuous, has Wiener law and is independent of $F_{S}$ by [F2], [F7] and coordinate generation. Random evaluation $(s, w) \mapsto w ((u_{i} - s)^{+})$ is measurable: replace its time by the ceiling on the dyadic mesh, use the countable coordinate formula, then pass to the pointwise limit by continuity. Hence the vectors $B_{u_{i}} = B_{u_{i} \land S} + W ((u_{i} - S)^{+})$ and ${\tilde{B}}_{u_{i}} = B_{u_{i} \land S} - W ((u_{i} - S)^{+})$ are product-measurable functions of $(R, W)$ . These formulas also hold when $S = T$ because $W (0) = 0$ . For every bounded Borel test of this vector, [F7] and invariance of Wiener measure under $w \mapsto - w$ from step 1.2 give equal conditional expectations. Thus the finite-dimensional laws agree, and [F5] gives equality on the entire cylinder sigma-algebra.

F1F2F5F7step 1.1step 1.2

3.1

On the event ${τ_{a} \leq T}$ , the reflected path coincides with $B$ on $[0, τ_{a}]$ , so $τ_{a} (\tilde{B}) = τ_{a}$ ; moreover ${\tilde{B}}_{T} = 2 a - B_{T}$ there. Hence ${τ_{a} \leq T, B_{T} \leq b} = {τ_{a} (\tilde{B}) \leq T, {\tilde{B}}_{T} \geq 2 a - b}$ as events. These are cylinder-measurable events on the continuous-path representatives: use the rational-distance formula of [F1] for the hit event and the terminal coordinate for the inequality. Thus by step 2.1 the probability equals $P (τ_{a} \leq T, B_{T} \geq 2 a - b)$ . Since $b \leq a$ implies $2 a - b \geq a$ , the event ${B_{T} \geq 2 a - b}$ is contained in ${B_{T} \geq a} \subseteq {τ_{a} \leq T}$ , so the last probability is $P (B_{T} \geq 2 a - b)$ . Together with $M_{T} \geq a ⟺ τ_{a} \leq T$ from step 1.1, this is the first identity of assertion 2.

F1step 1.1step 2.1

4.1

Taking $b = a$ in step 3.1 gives $P (M_{T} \geq a, B_{T} \leq a) = P (B_{T} \geq a)$ . Since ${B_{T} > a} \subseteq {M_{T} \geq a}$ and ${B_{T} > a}$ differs from ${B_{T} \geq a}$ by the null event ${B_{T} = a}$ by [F4], adding $P (B_{T} > a) = P (B_{T} \geq a)$ to both sides gives $P (M_{T} \geq a) = 2 P (B_{T} \geq a)$ , the second identity of assertion 2. Assertion 1 is step 2.1.

F4step 3.1

5.1

The endpoint and degenerate cases are covered: the statement requires $a > 0$ and $T > 0$ ; the value $b = a$ is used in step 4.1 and is the boundary case of the constraint $b \leq a$ ; the case $b < a$ is step 3.1 unchanged; the case $τ_{a} > T$ contributes to neither event in the first probability identity, by the containment in step 3.1; the case $τ_{a} \leq T$ with equality $B_{T} = a$ is the null event excluded by [F4]. AC is used through [F6] for the strong Markov and conditional-expectation interfaces.

F1F4F6givenstep 3.1∎

Source notes

Lawler, Proposition 2.7.2, derives the maximum formula from post-hit symmetry and absence of a terminal atom. Here the full reflected-path law is proved by conditioning on the bounded time tau_a wedge T, so the proof needs no prior almost-sure finiteness of tau_a. The measurable continuous-path representative is fixed before forming its filtration and hitting times.

CorollaryStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Law of the Brownian maximum

Statement

Assume the Axiom of Choice, let $B$ be a standard Brownian motion Brownian motion. Use the everywhere-continuous, zero-start representative fixed in Brownian reflection principle: replace the path by zero outside a measurable probability-one event of continuity and zero start, retaining the notation $B$ . Let $t > 0$ and $M_{t} : = \sup_{0 \leq s \leq t} B_{s}$ . This is a finite nonnegative random variable: continuity gives boundedness on $[0, t]$ and identifies its supremum with the supremum over the countable dense set $(Q \cap [0, t]) \cup {t}$ . With $Φ$ the standard normal distribution function, $Φ (x) = N (0, 1) ((- \infty, x])$ Standard normal and normal laws Cumulative distribution function of a real random variable, one has for every $x \geq 0$ $P (M_{t} \leq x) = 2 Φ ⁣ (\frac{x}{\sqrt{t}}) - 1.$ Consequently $M_{t}$ has the same law as $∣ B_{t} ∣$ , and on $x > 0$ the law of $M_{t}$ has the density $f (x) = \sqrt{\frac{2}{π t}} \exp ⁣ (- \frac{x^{2}}{2 t}) .$

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ in the stated everywhere-continuous zero-start representative, $t > 0$ and $x \geq 0$ .

[F1]

$P (M_{t} \geq a) = 2 P (B_{t} \geq a)$ for every $a > 0$ , and $M_{t} \geq a ⟺ \sup_{[0, t]} B \geq a$ . Brownian reflection principle

[F2]

The law of $B_{t}$ is $N (0, t)$ , the law of $\sqrt{t} Z$ for a standard normal $Z$ , whose density is $φ (y) = e^{- y^{2} / 2} / \sqrt{2 π}$ ; hence $P (a \leq B_{t} \leq b) = \int_{a}^{b} φ (y / \sqrt{t}) t^{- 1 / 2} d y$ for $a < b$ and $P (B_{t} = c) = 0$ . The Brownian kernels form a semigroup Standard normal and normal laws

[F3]

Substitution for continuous integrands on compact intervals, with the Riemann–Lebesgue equality (under countable choice, supplied by AC), and monotone convergence for increasing nonnegative functions, in particular indicators. Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral Monotone convergence for the integral

[F4]

A probability measure on $R$ is determined by its distribution function on the intervals $(- \infty, x]$ ; this uses countable choice, which AC supplies. Probability laws correspond to distribution functions The Axiom of Countable Choice ( ${A C}_{ω}$ ) The Axiom of Choice

Proof

technique · direct

1.1

For $x \geq 0$ and positive integers $n$ , put $a_{n} = x + 1 / n > 0$ . The events ${M_{t} \geq a_{n}}$ increase to ${M_{t} > x}$ , and ${B_{t} \geq a_{n}}$ increase to ${B_{t} > x}$ . Applying [F3] to their indicators and [F1] at each $a_{n}$ gives $P (M_{t} > x) = 2 P (B_{t} > x)$ . (To use an index starting at zero, replace $n$ by $n + 1$ .) By [F2], $P (B_{t} > x) = 1 - Φ (x / \sqrt{t})$ . Taking complements yields the asserted formula for every $x \geq 0$ , including $x = 0$ since the even normal density has mass one and no atom, so $Φ (0) = 1 / 2$ .

F1F2F3given

1.2

Define $G (x) : = \int_{0}^{x} f (y) d y$ for $x \geq 0$ with $f (y) = \sqrt{2 / (π t)} e^{- y^{2} / (2 t)}$ . The substitution $y = \sqrt{t} u$ , applied to the continuous integrand on $[0, x]$ , gives $G (x) = 2 (Φ (x / \sqrt{t}) - Φ (0)) = 2 Φ (x / \sqrt{t}) - 1$ for every $x > 0$ : indeed $f (\sqrt{t} u) \sqrt{t} = 2 φ (u)$ . For $x \geq 0$ , the bound $0 \leq G (x) \leq \sqrt{2 / (π t)} x$ gives $G (0^{+}) = 0$ , and the same computation with the upper limit tending to $+ \infty$ , together with $\lim_{u \to \infty} Φ (u) = 1$ , gives $\int_{0}^{\infty} f = 1$ .

F2F3

2.1

For $x \geq 0$ , $P (∣ B_{t} ∣ \leq x) = P (- x \leq B_{t} \leq x) = Φ (x / \sqrt{t}) - Φ (- x / \sqrt{t}) = 2 Φ (x / \sqrt{t}) - 1$ by the symmetry $Φ (- u) = 1 - Φ (u)$ of the standard normal law, which follows from the symmetry of its density $φ$ ; for $x < 0$ both $P (M_{t} \leq x)$ and $P (∣ B_{t} ∣ \leq x)$ vanish. Since the two distribution functions agree on all of $R$ , [F4] identifies the laws, so $M_{t}$ and $∣ B_{t} ∣$ have the same law.

F2F4step 1.1

2.2

The measure with density $f$ on $(0, \infty)$ , extended by zero on $(- \infty, 0]$ , is a probability measure whose distribution function at $x \geq 0$ is $G (x) = 2 Φ (x / \sqrt{t}) - 1$ and at $x < 0$ is $0$ ; by [F4] it therefore equals the law of $M_{t}$ . Hence the law of $M_{t}$ has the density $f$ on $x > 0$ and no atom at $0$ .

F4step 1.1step 1.2

3.1

The cases $x = 0$ and $t > 0$ are included in steps 1.1 and 2.2; the strict-tail identity was obtained by increasing indicator limits in step 1.1, and atomlessness of the maximum follows from its density in step 2.2. Countable choice in the Riemann–Lebesgue bridge and [F4] is supplied by the assumed AC.

F2F4givenstep 1.2∎

Source notes

Lawler, Proposition 2.7.2, supplies the reflection-based maximum formula. The strict tail is derived here by increasing indicator limits, including the endpoint zero. The density is identified through compact-interval substitution, the Riemann–Lebesgue bridge, and equality of distribution functions.

CorollaryStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Distribution of a one-sided Brownian hitting time

Statement

Assume the Axiom of Choice, let $B$ be a standard Brownian motion Brownian motion. Use the everywhere-continuous, zero-start representative of Law of the Brownian maximum: replace paths by zero outside a measurable probability-one event of continuity and zero start, retaining the notation $B$ . Let $a > 0$ and $τ_{a} : = \inf {t \geq 0 : B_{t} = a}$ , with $\inf \emptyset = + \infty$ . This is a measurable $[0, \infty]$ -valued hitting time for this representative; its distribution does not depend on the chosen full-measure event. With $Φ$ the standard normal distribution function Standard normal and normal laws Cumulative distribution function of a real random variable, $P (τ_{a} \leq t) = 2 (1 - Φ ⁣ (\frac{a}{\sqrt{t}})) (t > 0),$ and on $t > 0$ the law of $τ_{a}$ has the density $g (t) = \frac{a}{(2 π t^{3})^{1 / 2}} \exp ⁣ (- \frac{a^{2}}{2 t}) .$ Moreover $\lim_{t \to \infty} P (τ_{a} \leq t) = 1$ , so $τ_{a}$ is finite almost surely and there is no mass at infinity, and $P (τ_{a} = 0) = 0$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ in this everywhere-continuous zero-start representative, $a > 0$ and $t > 0$ .

[F1]

For the representative in the statement, $M_{t} = \sup_{[0, t]} B$ is a finite measurable random variable and $P (M_{t} \leq x) = 2 Φ (x / \sqrt{t}) - 1$ for $x \geq 0$ (Law of the Brownian maximum). The closed-set hitting-time lemma applies to an everywhere-continuous Brownian process with its own natural filtration (Brownian closed-set hitting times are stopping times). Brownian motion supplies a measurable full-measure continuity and zero-start event (Brownian motion).

[F2]

Limits of the standard normal distribution function: $\lim_{x \to + \infty} Φ (x) = 1$ , $Φ (0) = 1 / 2$ and $Φ$ is continuous, and $Φ (v) - Φ (u) = \int_{u}^{v} φ (x) d x$ for $u < v$ ; $φ (u) = e^{- u^{2} / 2} / \sqrt{2 π}$ . Standard normal and normal laws Cumulative distribution function of a real random variable The standard normal density has total mass one

[F3]

Substitution on compact intervals for the continuous integrand $2 φ$ , and monotone convergence for the limits at the endpoints. Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ Monotone convergence for the integral. The compact integrals agree with their Lebesgue counterparts under Countable Choice. A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral

[F4]

A probability measure on $R$ is determined by its distribution function; countable choice, which AC supplies, is used there. Probability laws correspond to distribution functions The Axiom of Countable Choice ( ${A C}_{ω}$ ) The Axiom of Choice

[F5]

Closed bounded real intervals are compact, a continuous function attains its maximum on a nonempty compact set, and the intermediate value theorem holds on real intervals (Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact, Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value, Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ ). A nonnegative measurable density defines a measure by integration (The indefinite integral of a nonnegative measurable function is a measure).

Proof

technique · direct

1.1

Fix the measurable full-measure event $A$ specified in the statement. Replacing the original path by zero on $A^{c}$ preserves every finite-dimensional law and makes every path continuous with $B_{0} = 0$ . Each coordinate remains measurable because $A$ is measurable. Two such choices agree on the intersection of their events, so the resulting measurable hitting times agree there and have the same distribution. By [F1] applied to the closed singleton ${a}$ , $τ_{a}$ is a stopping time for the chosen process's own raw natural filtration, hence an extended nonnegative measurable random variable. No stopping-time claim for the original raw filtration is used.

F1given

1.2

Define $g (s) = a (2 π s^{3})^{- 1 / 2} e^{- a^{2} / (2 s)}$ for $s > 0$ . For $0 < ε < t$ , the substitution $u = a / \sqrt{s}$ on $[ε, t]$ , whose derivative $- a / (2 s^{3 / 2})$ is continuous and $2 φ$ is continuous, gives $\int_{ε}^{t} g (s) d s = \int_{a / \sqrt{t}}^{a / \sqrt{ε}} 2 φ (u) d u = 2 (Φ (a / \sqrt{ε}) - Φ (a / \sqrt{t}))$ by oriented substitution, the compact Riemann/Lebesgue bridge, and the density-integral identity for increments of $Φ$ .

F2F3

2.1

For $t > 0$ , if $τ_{a} \leq t$ , the first hit is attained by continuity (as in the closed-set hitting lemma), so $M_{t} \geq a$ . Conversely $M_{t} \geq a$ gives a time $s \in [0, t]$ with $B_{s} = M_{t}$ by [F5]; since $B_{0} = 0 < a \leq B_{s}$ , the intermediate value theorem gives a hit by time $s$ . Hence ${τ_{a} \leq t} = {M_{t} \geq a}$ as exact measurable events for this representative. Continuity at zero also gives $τ_{a} > 0$ on every path, since $B_{0} = 0 < a$ .

F1F5step 1.1

3.1

The normal CDF obeys $∣ Φ (v) - Φ (u) ∣ \leq ∣ v - u ∣ / \sqrt{2 π}$ , by its density bound, hence is continuous. Symmetry and total mass one give $Φ (0) = 1 / 2$ ; monotone convergence of density integrals gives $Φ (x) \to 1$ as $x \to \infty$ . Put $x_{n} = a (1 - 1 / n)$ for $n \geq 1$ . The measurable events ${M_{t} \leq x_{n}}$ increase to ${M_{t} < a}$ , so [F3] and [F1] give $P (M_{t} < a) = \lim_{n} (2 Φ (x_{n} / \sqrt{t}) - 1) = 2 Φ (a / \sqrt{t}) - 1 = P (M_{t} \leq a)$ . Thus $P (M_{t} = a) = 0$ , and step 2.1 yields $P (τ_{a} \leq t) = 2 (1 - Φ (a / \sqrt{t}))$ . Taking integer $t \to \infty$ and monotone convergence of the events ${τ_{a} \leq t}$ gives $P (τ_{a} < \infty) = 1$ .

F1F2F3step 2.1

4.1

Let $ε = t / n$ in step 1.2 and let integers $n \geq 2$ tend to infinity. The nonnegative integrals increase to $\int_{0}^{t} g (s) d s$ , and $Φ (a / \sqrt{ε}) \to 1$ , so $\int_{0}^{t} g (s) d s = 2 (1 - Φ (a / \sqrt{t})) = P (τ_{a} \leq t)$ . Letting integer $t \to \infty$ now gives $\int_{0}^{\infty} g = 1$ .

F2F3step 3.1step 1.2

5.1

Extend $g$ by zero on $(- \infty, 0]$ . It is nonnegative Borel measurable, and [F5] and step 4.1 make its density measure a Borel probability measure on $R$ . To use [F4] with a real random variable, replace $τ_{a} = \infty$ by the value $1$ on its measurable null event, obtaining ${\tilde{τ}}_{a}$ . This leaves every finite-time distribution probability unchanged, and ${\tilde{τ}}_{a} > 0$ by step 2.1. The density measure and ${\tilde{τ}}_{a}$ have CDF zero for nonpositive arguments, and the same CDF at every positive argument by step 4.1. Thus [F4] identifies the laws. In particular the original extended hitting time has density $g$ on $(0, \infty)$ , no atom there or at zero, and no mass at infinity.

F4F5step 2.1step 3.1step 4.1

6.1

The parameter $a > 0$ and compact substitution bounds $0 < ε < t$ ensure every denominator is positive. At $t = 0$ , step 2.1 gives $P (τ_{a} = 0) = 0$ . The infinity limit and total density mass were proved in steps 3.1 and 4.1. AC supplies the Countable Choice hypotheses of both the compact integration bridge and [F4], and the Brownian and hitting-time suppliers. The event equality uses the declared continuous representative throughout.

F1F3F4step 2.1step 3.1step 4.1step 5.1∎

The proof combines the Brownian maximum law with an exact continuous-path hitting identity. It computes the density integral by compact substitution, the Riemann/Lebesgue bridge and monotone limits, then identifies probability laws through their CDFs on all real arguments.

CorollaryStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

One-dimensional Brownian motion hits every point almost surely

Statement

Assume the Axiom of Choice and let $B$ be a standard Brownian motion Brownian motion. Use the everywhere-continuous, zero-start representative fixed in Distribution of a one-sided Brownian hitting time: replace the path by zero outside a measurable probability-one event of continuity and zero start, retaining the notation $B$ . For $a \in R$ , let $τ_{a} = \inf {t \geq 0 : B_{t} = a}$ , with $\inf \emptyset = + \infty$ . Then each $τ_{a}$ is a measurable $[0, \infty]$ -valued hitting time and $P (τ_{a} < \infty) = 1$ for every $a \in R$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ in the stated everywhere-continuous zero-start representative, and $a \in R$ .

[F1]

For the representative in the statement and $a > 0$ , $τ_{a}$ is a measurable extended random variable and, for $t > 0$ , $P (τ_{a} \leq t) = 2 (1 - Φ (a / \sqrt{t}))$ ; the right side tends to $1$ as $t \to \infty$ because $Φ$ is continuous at $0$ with $Φ (0) = 1 / 2$ . Distribution of a one-sided Brownian hitting time Standard normal and normal laws Cumulative distribution function of a real random variable

[F2]

Probability measures are continuous from below along increasing sequences of events. Basic identities for a probability measure

[F3]

If $B$ is a standard Brownian motion then so is $- B$ : $- B_{0} = 0$ almost surely, the increments change sign and centered normal laws are symmetric, and continuity is unchanged. Moreover $τ_{a} (B) = τ_{- a} (- B)$ pathwise, because $B_{t} = a$ if and only if $- B_{t} = - a$ . Brownian motion

[F4]

The chosen representative satisfies $B_{0} = 0$ on every outcome, so $τ_{0} = 0$ everywhere. Distribution of a one-sided Brownian hitting time

[F5]

AC is the standing hypothesis under which the Brownian and hitting-time interfaces in [F1], [F3] and [F4] are supplied; no additional path is selected here. The Axiom of Choice

Proof

technique · direct

1.1

Let $a > 0$ . The events ${τ_{a} \leq t}$ increase with $t$ to ${τ_{a} < \infty}$ , so [F2] applied to the sequence $t = n + 1$ gives $P (τ_{a} < \infty) = \lim_{n \to \infty} P (τ_{a} \leq n + 1) = \lim_{n \to \infty} 2 (1 - Φ (a / \sqrt{n + 1})) = 2 (1 - Φ (0)) = 1$ by [F1].

F1F2given

1.2

For $a = 0$ the identity $τ_{0} = 0$ holds everywhere by [F4], so $τ_{0}$ is measurable and $P (τ_{0} < \infty) = 1$ .

F4given

2.1

Let $a < 0$ . By [F3] the process $- B$ is a standard Brownian motion in an everywhere-continuous zero-start representative and $τ_{a} (B) = τ_{- a} (- B)$ pathwise with $- a > 0$ ; [F1] makes the latter hitting time measurable, and step 1.1 applied to $- B$ and the level $- a$ gives $P (τ_{a} (B) < \infty) = P (τ_{- a} (- B) < \infty) = 1$ .

F1F3step 1.1

3.1

The cases $a > 0$ , $a = 0$ and $a < 0$ are exhaustive, so $P (τ_{a} < \infty) = 1$ for every real $a$ . The conclusion concerns the first hitting time only; it does not assert finiteness of the expectation, and the case of a level already occupied at time $0$ is contained in the $a = 0$ case while for $a \neq 0$ the start $B_{0} = 0$ is a.s. distinct from $a$ . AC is used only through [F5].

F5givenstep 1.1step 1.2step 2.1∎

Source notes

Durrett, Section 7.4, reads the almost-sure finiteness off the first-passage distribution at $t \to \infty$ ; Sousi, Section 6.7, uses the same consequence for recurrence. The symmetry step is proved from the Brownian definition itself, so no separate invariance theorem for Wiener measure is assumed.

DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-09-22Open item page →

Brownian motion started at x

Definition

Assume the Axiom of Choice. Let $d \geq 1$ be a finite integer and let $B$ be a standard $d$ -dimensional Brownian motion on a probability space $(Ω, F, P)$ $d$ -dimensional Brownian motion, as supplied by Existence and scaling of $d$ -dimensional Brownian motion. For $x \in R^{d}$ , choose the measurable probability-one event on which $B_{0} = 0$ and the path $t \mapsto B_{t}$ is continuous, and put $\hat{B} = B$ on that event and ${\hat{B}}_{t} = 0$ for every $t$ off it. Then put $B_{t}^{x} : = x + {\hat{B}}_{t}, t \geq 0.$ The process $B^{x}$ is the standard $d$ -dimensional Brownian motion started at $x$ , and its law $P_{x} : = the law of the random element ω \mapsto (x + {\hat{B}}_{t} (ω))_{t \geq 0}$ is called the shifted Brownian law at $x$ . Here the target is the canonical continuous path space $C ([0, \infty), R^{d})$ with its compact-open Borel sigma-algebra. The finite-dimensional version of Borel sigma-algebra of continuous path space is generated by coordinates (applied coordinatewise) says that this Borel sigma-algebra is generated by the evaluations $f \mapsto f (t)$ . Thus the displayed map is a random element, and $P_{x}$ is a probability measure The law of a random element is a probability measure. We write $P_{x} (A) = P ((x + {\hat{B}}_{t})_{t \geq 0} \in A)$ .

The following are part of the definition and are used later in this form.

Initial value and path space. Every path in the image is continuous and starts at $x$ . In particular $P_{x} (f (0) = x) = 1$ . The normalization changes $B$ only on a null event and therefore changes none of its finite-dimensional distributions.
Increments. For $0 \leq s \leq t$ one has the pathwise identity $B_{t}^{x} - B_{s}^{x} = {\hat{B}}_{t} - {\hat{B}}_{s}$ . Consequently, under $P_{x}$ the increments are independent with laws $N_{d} (0, (t - s) I_{d})$ $d$ -dimensional Brownian motion; in particular $B^{x}$ is again a standard Brownian motion up to its initial value $x$ .
Translation of hitting times. Let $C \subseteq R^{d}$ be closed and let $T_{C} (f) : = \inf {t \geq 0 : f (t) \in C}$ (with $\inf \emptyset : = + \infty$ ) be the first hitting functional of $C$ , evaluated on path space. Then pathwise $T_{C} (B^{x}) = T_{C - x} (\hat{B}), C - x : = {z - x : z \in C}$ because $x + {\hat{B}}_{t} \in C$ if and only if ${\hat{B}}_{t} \in C - x$ . The functional $T_{C}$ is Borel on continuous path space: for finite $t$ , ${T_{C} \leq t}$ is the closed set of paths whose compact restriction to $[0, t]$ meets $C$ . In particular $P_{x} (T_{C} \leq t) = P (T_{C - x} \leq t)$ for every $t \geq 0$ , and for $d = 1$ the one-point case reads $P_{x} (T_{a} < \infty) = P (T_{a - x} < \infty)$ .
These are the only shifted laws used below. The one-dimensional items use $d = 1$ , the planar items use $d = 2$ , and $P_{0}$ is the law of $B$ itself. No statement below treats $P_{x}$ as a kernel in $x$ or as a regular conditional distribution.

Source notes

Durrett, Section 7.5, and Sousi, Section 6.1, use the notation $P_{x}$ for Brownian motion started at $x$ without minting a separate definition. The definition above fixes that notation on canonical continuous path space, so that closed-set hitting-time events are Borel events of the shifted law, and records the translation identity that every later use consumes.

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Two-sided Brownian exit probability

Statement

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion Brownian motion, let $a < x < b$ be reals, and let $P_{x}$ be the law of the shifted, everywhere-continuous process $t \mapsto x + {\hat{B}}_{t}$ Brownian motion started at x, on canonical continuous path space with its compact-open Borel sigma-algebra. Here $\hat{B}$ is the normalized zero-start representative specified in that definition. For $c \in R$ let $T_{c} : = \inf {t \geq 0 : Z_{t} = c}$ be the hitting time of the level $c$ for the coordinate process $Z$ of the shifted law. Then $P_{x} (T_{b} < T_{a}) = \frac{x - a}{b - a} .$

Facts & Assumptions

Given: AC, a standard Brownian motion, reals $a < x < b$ , and the shifted law $P_{x}$ . In the proof write $B$ for its everywhere-continuous zero-start representative $\hat{B}$ and use that representative's own raw natural filtration and usual augmentation. No adaptation to a former raw filtration is claimed.

[F1]

For the shifted law $P_{x}$ , hitting times satisfy $P_{x} (T_{c} < \infty) = P (T_{c - x} < \infty)$ and $P_{x} (T_{b} < T_{a}) = P (T_{b - x} < T_{a - x})$ , because the shifted process is $x + B$ . Brownian motion started at x

[F2]

One-dimensional Brownian motion hits every level almost surely: $P (T_{c} < \infty) = 1$ for every $c$ . One-dimensional Brownian motion hits every point almost surely

[F3]

The hitting time of a closed set for the normalized everywhere-continuous $B$ is a stopping time for its raw natural filtration and its usual augmentation. The maximum $M_{N} = \sup_{s \leq N} B_{s}$ has the same law as $∣ B_{N} ∣$ for $N > 0$ . Hence $E M_{N} = E ∣ B_{N} ∣ \leq \sqrt{E B_{N}^{2}} = \sqrt{N}$ . The same argument applies to $- B$ , which is Brownian by symmetry of its Gaussian increments. Since $B_{0} = 0$ , $\sup_{s \leq N} ∣ B_{s} ∣ \leq \sup_{s \leq N} B_{s} + \sup_{s \leq N} (- B_{s})$ , whose expectation is at most $2 \sqrt{N}$ . The suprema are measurable rational-time suprema by continuity; $N = 0$ gives zero directly. Brownian closed-set hitting times are stopping times Law of the Brownian maximum Standard normal and normal laws Cauchy-Schwarz for random variables Brownian motion

[F4]

Strong Markov: for an a.s. finite stopping time $τ$ of the usual augmentation, the increment process $(B_{τ + t} - B_{τ})_{t \geq 0}$ is a Brownian motion independent of $F_{τ}$ . Strong Markov property of Brownian motion Continuous-time stopping times and stopped sigma-algebras Natural and usual augmented Brownian filtrations

[F5]

The law of $B_{M}$ is $N (0, M)$ , which is centered and has no atoms; the tower identity gives $E [E [X ∣ G]] = E X$ . The Brownian kernels form a semigroup Standard normal and normal laws Basic algebra and order properties of conditional expectation Conditional expectation as an ae class Conditional expectation is unique almost surely

[F6]

Dominated convergence and monotone convergence pass limits through integrals. Dominated convergence Monotone convergence for the integral

[F7]

AC supplies the conditional-expectation and strong-Markov interfaces. The Axiom of Choice Wiener measure on continuous path space

Proof

technique · direct

1.1

By [F1] it suffices to prove the case $a < 0 < b$ with the unshifted law: $P_{x} (T_{b} < T_{a}) = P (T_{b - x} < T_{a - x})$ and $(x - a) / (b - a) = (0 - (a - x)) / ((b - x) - (a - x))$ , since $(b - x) - (a - x) = b - a$ and $a - x < 0 < b - x$ . So assume $a = A < 0 < B = b$ and put $T : = T_{A} \land T_{B}$ . Then $T \leq T_{A} < \infty$ almost surely by [F2], and $T$ is a stopping time of the usual augmentation because ${A, B}$ is closed, by [F3]. For every $u < T$ the path satisfies $B_{u} \in (A, B)$ , since leaving $(A, B)$ would require hitting $A$ or $B$ by continuity.

F1F2F3given

1.2

By [F4] applied at the a.s. finite stopping time $T$ , the process $W_{t} = B_{T + t} - B_{T}$ on ${T < \infty}$ and $W_{t} = 0$ otherwise is an everywhere-continuous zero-start Brownian motion independent of $F_{T}$ , using the supplier's measurable random-time convention. Consequently, for every $F_{T}$ -measurable random variable $R$ with values in $[0, N]$ one has $E [W_{R} ∣ F_{T}] = 0$ almost surely. Indeed, if $R$ is countably valued with values $r_{j} \in [0, N]$ , then $W_{R} = \sum_{j} W_{r_{j}} 1_{{R = r_{j}}}$ and for $G \in F_{T}$ one has $\int_{G} W_{R} d P = \sum_{j} P (G \cap {R = r_{j}}) E W_{r_{j}} = 0$ , because $G \cap {R = r_{j}} \in F_{T}$ is independent of $W_{r_{j}}$ and $E W_{r_{j}} = 0$ ; for general $R$ the dyadic ceilings $R_{k} : = \min (2^{- k} ⌈ 2^{k} R ⌉, N)$ decrease to $R$ , so $W_{R_{k}} \to W_{R}$ almost surely and $∣ W_{R_{k}} ∣ \leq S : = \sup_{[0, N]} ∣ W ∣$ , whose expectation is finite by [F3]; dominated convergence [F6] gives $\int_{G} W_{R} d P = \lim_{k} \int_{G} W_{R_{k}} d P = 0$ for every $G \in F_{T}$ , and [F5]'s uniqueness identifies $E [W_{R} ∣ F_{T}] = 0$ .

F3F4F5F6

2.1

Fix $M > \max {∣ A ∣, ∣ B ∣}$ and let $R = M - T$ on ${T \leq M}$ and $R = 0$ otherwise. This is an $F_{T}$ -measurable random variable with values in $[0, M]$ : $T$ is $F_{T}$ -measurable since ${T \leq r} \cap {T \leq u} = {T \leq \min (r, u)} \in F_{u}$ . No product $0 \cdot \infty$ is used. On ${T \leq M}$ one has $B_{M} - B_{T} = W_{M - T} = W_{R}$ , and on ${T > M}$ both $B_{M} - B_{T \land M}$ and $W_{R} = W_{0}$ vanish; hence $B_{M} - B_{T \land M} = W_{R}$ . All terms are integrable: $B_{M}$ is Gaussian, $W_{R}$ is bounded in absolute value by its integrable finite-horizon supremum, and the identity gives integrability of $B_{T \land M}$ . Taking expectations and using step 1.2 with [F5]'s tower identity, $E [B_{M} - B_{T \land M}] = 0$ , so $E [B_{T \land M}] = E [B_{M}] = 0$ , the last equality because the law of $B_{M}$ is the centered $N (0, M)$ .

F5step 1.2

3.1

Let $M \to \infty$ . Set $B_{T} = 0$ on the null event $T = \infty$ , as in the strong-Markov convention. The random variables $B_{T \land M}$ converge almost surely to $B_{T}$ because $T < \infty$ almost surely and the paths are continuous, and they are bounded by $\max {∣ A ∣, ∣ B ∣}$ : for $t < T$ the value $B_{t}$ lies in $(A, B)$ by step 1.1, and $B_{T} \in {A, B}$ . Dominated convergence [F6] therefore gives $E [B_{T}] = 0$ .

F6step 1.1step 2.1

4.1

The events ${T_{B} < T_{A}}$ and ${T_{A} < T_{B}}$ are disjoint and their union is almost surely the whole space, because $T < \infty$ almost surely and $T_{A} \neq T_{B}$ almost surely (the path cannot be at two distinct levels at one time). On the first event $B_{T} = B$ and on the second $B_{T} = A$ , so $0 = E [B_{T}] = B P (T_{B} < T_{A}) + A (1 - P (T_{B} < T_{A}))$ ; solving gives $P (T_{B} < T_{A}) = - A / (B - A)$ .

step 3.1

5.1

Undoing the shift with step 1.1, $P_{x} (T_{b} < T_{a}) = P (T_{b - x} < T_{a - x}) = \frac{- (a - x)}{(b - x) - (a - x)} = \frac{x - a}{b - a}$ , which is the assertion.

step 1.1step 4.1

6.1

The endpoint cases are covered: the strict inequalities $a < x < b$ keep $T_{a}$ and $T_{b}$ distinct from the starting level; the truncation parameter $M$ is chosen larger than both endpoints and then sent to infinity in step 3.1; the case $A = B$ is excluded because $a < b$ ; and $T_{A} = T_{B}$ is the null event excluded in step 4.1. AC is used only through [F7] in the conditional-expectation and strong-Markov interfaces.

F4F7givenstep 4.1∎

Source notes

Durrett, Theorem 7.5.3, proves the complementary lower-exit formula by bounded stopping and bounded convergence. Solving its endpoint expectation identity gives the stated upper-exit formula. The proof above instead verifies the centered martingale identity $E [B_{T}] = 0$ through the strong Markov restart at $T$ , which keeps every step within the stopping-time and maximum machinery already established on this page.

CorollaryStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

One-dimensional Brownian motion is recurrent

Statement

Assume the Axiom of Choice and let $B$ be a standard Brownian motion Brownian motion. Use the everywhere-continuous, zero-start representative fixed in One-dimensional Brownian motion hits every point almost surely, retaining the notation $B$ . Then almost surely the set ${t \geq 0 : B_{t} \in I}$ is unbounded for every nonempty open interval $I \subseteq R$ ; equivalently, almost surely the path visits every neighbourhood of every real point at arbitrarily large times.

Facts & Assumptions

Given: AC and a standard Brownian motion $B$ in the stated fixed everywhere-continuous, zero-start representative.

[F1]

For this representative, one-dimensional Brownian motion hits every deterministic level almost surely: $P (T_{c} < \infty) = 1$ for every $c \in R$ . One-dimensional Brownian motion hits every point almost surely

[F2]

Future-path Markov: for each deterministic $s \geq 0$ and each bounded Borel functional $ϕ$ on continuous path space, $E [ϕ ((B_{s + t})_{t \geq 0}) ∣ F_{s}] = \int ϕ (B_{s} + w) μ (d w)$ almost surely, where $μ$ is Wiener measure. Future-path Markov property Natural and usual augmented Brownian filtrations

[F3]

Conditional-expectation versions are unique almost surely, so an event whose conditional probability given $F_{s}$ equals $1$ has probability one. Conditional expectation as an ae class Conditional expectation is unique almost surely

[F4]

Countable intersections of probability-one events have probability one, by continuity from above of a probability measure based at a probability-one event. Basic identities for a probability measure

[F5]

The rationals are dense in $R$ , so every nonempty open interval contains a rational point. The rationals embed densely in the reals

[F6]

AC is the ambient assumption of the Brownian construction. The Axiom of Choice

Proof

technique · direct

1.1

Fix $c \in R$ and define on continuous path space $ϕ_{c} (v) = 1_{{\exists t \geq 0 : v (t) = c}}$ . This functional is Borel: its one-set is $⋃_{m \geq 1} {v : \min_{0 \leq t \leq m} ∣ v (t) - c ∣ = 0}$ , and each displayed minimum is continuous for uniform convergence on $[0, m]$ (changing the path by at most $ε$ changes the minimum by at most $ε$ ). For every deterministic $x \in R$ , [F1] applied under Wiener measure to the level $c - x$ gives $\int ϕ_{c} (x + w) μ (d w) = 1$ .

2.1

Fix $n \geq 1$ and let $A_{c, n} : = {\exists t \geq n : B_{t} = c}$ . Because the chosen representative is everywhere continuous, $1_{A_{c, n}} = ϕ_{c} ((B_{n + t})_{t \geq 0})$ pointwise. Applying [F2] and step 1.1 at time $n$ therefore gives $P (A_{c, n} ∣ F_{n}) = 1$ almost surely; by [F3] this forces $P (A_{c, n}) = 1$ . This conditions a fixed Borel future-path event and evaluates its kernel at the known state $B_{n}$ ; it does not apply [F1] directly to a random level.

F1F2F3givenstep 1.1

3.1

For fixed $c$ the events $A_{c, 1} \supseteq A_{c, 2} \supseteq \dots$ all have probability one, so $A_{c} : = ⋂_{n \geq 1} A_{c, n}$ has probability one by [F4], and on $A_{c}$ the path visits the level $c$ at arbitrarily large times.

F4step 2.1

4.1

The intersection $A : = ⋂_{c \in Q} A_{c}$ over the countable set of rationals again has probability one by [F4]; on $A$ , for every rational $c$ and every time bound the path visits $c$ at some larger time.

F4step 3.1

5.1

Let $I \subseteq R$ be a nonempty open interval. By [F5] choose a rational $c \in I$ . On the probability-one event $A$ of step 4.1 the path visits $c$ , hence enters $I$ , at arbitrarily large times. Since every nonempty open interval arises in this way and $A$ does not depend on $I$ , almost surely the set ${t : B_{t} \in I}$ is unbounded for every nonempty open interval $I$ .

F5step 4.1

6.1

The equivalent formulation follows: for a real point $y$ and $ε > 0$ , the interval $(y - ε, y + ε)$ is nonempty and open, so it is visited at arbitrarily large times almost surely. The case of the empty interval is excluded, singleton intervals are not claimed as infinitely visited except through the containing open intervals, and the conclusion is about the unboundedness of the visit set, not about any integrability of a hitting time; the first visit of a fixed level is the almost-sure finiteness proved in [F1]. AC is used only through [F6].

F1F6givenstep 5.1∎

Source notes

On the source side, Sousi, Section 6.7, proves one-dimensional recurrence from the almost-sure finiteness of hitting times together with the restart argument, and Durrett, Section 7.4, records the same consequence. The statement here is the neighbourhood form actually consumed by the planar example on the companion page, which contrasts it with the polarity of single points in the plane.

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-22Open item page →

Planar Brownian annular exit probability

Statement

Assume the Axiom of Choice. Let $P_{x}$ be the shifted planar Brownian law on canonical continuous path space Brownian motion started at x, and let $Z$ be its coordinate process. For $x, y \in R^{2}$ with $x \neq y$ and $0 < ε < ∣ x - y ∣ < R$ , let $S_{ε} : = \inf {t \geq 0 : ∣ Z_{t} - y ∣ = ε}, T_{R} : = \inf {t \geq 0 : ∣ Z_{t} - y ∣ = R},$ and let $H : = \inf {t \geq 0 : ∣ Z_{t} - y ∣ \notin (ε, R)}$ . Then $P_{x} (S_{ε} < T_{R}) = \frac{\log R - \log ∣ x - y ∣}{\log R - \log ε} .$

Facts & Assumptions

Given: AC, the continuous coordinate process $Z$ under $P_{x}$ , $x \neq y$ in $R^{2}$ and $0 < ε < ∣ x - y ∣ < R$ .

[F1]

Under $P_{x}$ , $Z_{0} = x$ almost surely, the increments of $Z$ over $[s, t]$ have law $N_{2} (0, (t - s) I_{2})$ and are independent of the raw coordinate past, and each process $Z^{i} - Z_{0}^{i}$ is a standard one-dimensional Brownian motion. Every coordinate path is continuous. $d$ -dimensional Brownian motion Brownian motion started at x

[F2]

One-dimensional Brownian motion hits every level almost surely. The stopping-time definition uses exact events; the required event identities are proved in step 1.1. The conditioning lemma gives $E [h (X, Y) ∣ G] = H (X)$ for a known state $X$ and independent noise $Y$ , $H (x) = \int h (x, y) μ (d y)$ . One-dimensional Brownian motion hits every point almost surely Conditioning a known state and independent noise Continuous-time stopping times and stopped sigma-algebras

[F3]

$N_{2} (0, I_{2})$ is the law of a pair of independent standard normal coordinates, whose one-dimensional density $φ$ is positive with $\int φ = 1$ and finite second moment. In particular $E ∣ G_{i} ∣ < \infty$ follows from $∣ u ∣ \leq 1 + u^{2}$ . Gaussian even moments for Brownian increments Multivariate normal law, including singular covariance Standard normal and normal laws

[F4]

Compact integration by parts, the chain rule, the fundamental theorem of calculus for a differentiable primitive, dominated and monotone convergence, and Tonelli/Fubini for bounded or integrable product integrands. If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ Dominated convergence Monotone convergence for the integral Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Fubini's theorem for L^1 functions on a sigma-finite product If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative

[F8]

Continuous real functions on compact intervals attain their extrema and take all intermediate values. The mean value theorem bounds difference quotients. Under AC, the Countable Choice Riemann-to-Lebesgue bridge identifies the compact calculus integrals in [F4] with Lebesgue integrals. Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral

[F5]

Optional sampling for bounded discrete stopping times: for a martingale $Y$ with $E [Y_{k + 1} ∣ G_{k}] = Y_{k}$ and stopping times $0 \leq σ \leq τ$ bounded by $N$ , $E [Y_{τ}] = E [Y_{σ}]$ ; a discrete martingale is defined by its adjacent conditional means, and the discrete stopped sigma-algebra is defined by the events ${τ \leq k}$ . Martingale submartingale and supermartingale Optional sampling for bounded stopping times Continuous-time filtrations and all-pairs martingales

[F6]

Conditional expectations are unique almost surely and are additive on bounded inputs. Conditional expectation as an ae class Conditional expectation is unique almost surely Basic algebra and order properties of conditional expectation Measurability of integration against a kernel Measure kernel and probability kernel

[F7]

Full AC supplies the Brownian and conditional-expectation interfaces and the inherited Countable Choice in the compact Riemann-to-Lebesgue bridge. The Axiom of Choice Brownian motion

Proof

technique · direct

1.1

Put $a_{s} = ∣ Z_{s} - y ∣$ . Every path is continuous. For $t \geq 0$ , compact attainment and rational approximation give ${H \leq t} = ⋂_{m \geq 1} ⋃_{q \in (Q \cap [0, t]) \cup {t}} {a_{q} < ε + 1 / m or a_{q} > R - 1 / m} .$ The reverse inclusion follows since the continuous nonnegative distance of $a_{s}$ to the closed set $(- \infty, ε] \cup [R, \infty)$ then has minimum zero on $[0, t]$ . Similarly, for $c = ε$ or $R$ , its circle hitting time has event $⋂_{m \geq 1} ⋃_{q \in (Q \cap [0, t]) \cup {t}} {∣ a_{q} - c ∣ < 1 / m} .$ These countable events are in the raw coordinate past, so all three times are stopping times and their comparisons are measurable. On the common probability-one event $Z_{0} = x$ , the initial radius is strictly between the boundaries. Continuity and the intermediate value theorem give $H = S_{ε} \land T_{R}$ , the boundary value $∣ Z_{H} - y ∣ \in {ε, R}$ when $H < \infty$ , and $∣ Z_{s} - y ∣ \in (ε, R)$ for $s < H$ . These last claims are used only on that event.

F1F2F8given

1.2

Define $ψ (r) : = \log r$ for $r \in [ε, R]$ and extend it to a $C^{2}$ function on $[0, \infty)$ with $ψ \equiv \log ε$ on $[0, ε / 2]$ , $ψ \equiv \log R$ on $[2 R, \infty)$ , and quintic Hermite splices on $[ε / 2, ε]$ and $[R, 2 R]$ that match value, first and second derivative at both joints: on $[ε / 2, ε]$ use $s \mapsto \log ε + (\log s - \log ε) h (2 s / ε - 1)$ and on $[R, 2 R]$ use $s \mapsto \log R + (\log s - \log R) (1 - h (s / R - 1))$ , where $h (θ) = 6 θ^{5} - 15 θ^{4} + 10 θ^{3}$ satisfies $h = h^{'} = h^{''} = 0$ at $θ = 0$ and $h = 1$ , $h^{'} = h^{''} = 0$ at $θ = 1$ . Each splice agrees with the neighbouring branches in value and in its first two derivatives at both endpoints, so the resulting $ψ$ is $C^{2}$ with bounded first and second derivatives, and $Φ (z) : = ψ (∣ z - y ∣)$ is then a bounded $C^{2}$ function of $z \in R^{2}$ , constant near $y$ and outside the disc of radius $2 R$ , with $Φ (z) = \log ∣ z - y ∣$ for $∣ z - y ∣ \in [ε, R]$ ; its Laplacian $Δ Φ (z) = ψ^{''} (r) + ψ^{'} (r) / r$ at $r = ∣ z - y ∣ > 0$ is continuous and bounded, and it vanishes on ${ε \leq ∣ z - y ∣ \leq R}$ because $Δ \log ∣ z - y ∣ = 0$ there.

F4algebra

1.3

For the standard normal pair $G = (G_{1}, G_{2})$ of [F3], every bounded $C^{2}$ function $f$ with bounded derivatives satisfies $E [\partial_{i} f (z + \sqrt{r} G) G_{i}] = \sqrt{r} E [\partial_{i}^{2} f (z + \sqrt{r} G)]$ for $i = 1, 2$ and $r > 0$ . Indeed, the law of $G$ is the product of the two standard normal laws by [F3], so Fubini expresses the expectation as an iterated integral. Fix the other coordinate and integrate by parts in the chosen one-dimensional coordinate on $[- L, L]$ with the compact theorem of [F4] using $φ^{'} = - u φ$ and the bounded factor $u \mapsto \partial_{i} f (z + \sqrt{r} (u e_{i} + v e_{j}))$ , where $j \neq i$ and the other Gaussian coordinate $v$ is fixed; the boundary terms vanish as $L \to \infty$ because $φ$ decays rapidly and the derivative factor is bounded, and dominated convergence identifies the limit. On each finite interval the integrands are continuous, so [F8] identifies the compact integration-by-parts identity with its Lebesgue version. The bounds are independent of the fixed other coordinate; Fubini completes that coordinate integration. Summing the two coordinates gives $E [\nabla f (z + \sqrt{r} G) \cdot G] = \sqrt{r} E [Δ f (z + \sqrt{r} G)]$ .

F3F4F8

1.4

Let $G_{s} : = σ (Z_{u} : 0 \leq u \leq s)$ be the raw coordinate filtration. For every bounded Borel $g : R^{2} \to R$ and $0 \leq s \leq t$ one has $E_{x} [g (Z_{t}) ∣ G_{s}] = Q_{t - s} g (Z_{s})$ almost surely: apply the conditioning lemma to the known state $Z_{s}$ and independent noise $Z_{t} - Z_{s}$ .

F1F2given

2.1

The standard one-dimensional Brownian motion $Z^{(1)} - x_{1}$ (zero-start almost surely) hits the level $y_{1} + R - x_{1}$ almost surely. At that time $∣ Z_{t} - y ∣ \geq ∣ Z_{t}^{(1)} - y_{1} ∣ = R$ , so $H$ is no larger and is finite $P_{x}$ -almost surely.

F1F2step 1.1

2.2

Fix a bounded $C^{2}$ function $f$ with bounded first and second derivatives and put $Q_{r} f (z) : = E [f (z + \sqrt{r} G)] = \int f (z + u) μ_{r} (d u)$ with $μ_{r}$ the law of $\sqrt{r} G$ . Then $r \mapsto Q_{r} f (z)$ is differentiable on $(0, \infty)$ with $\frac{d}{d r} Q_{r} f (z) = \frac{1}{2} Q_{r} Δ f (z)$ : differentiating the expectation is licensed by the mean value theorem in [F8] and dominated convergence on a neighborhood bounded away from $r = 0$ , because $\nabla f$ is bounded and $∣ G ∣$ is integrable, and the resulting expression $E [\nabla f (z + \sqrt{r} G) \cdot G / (2 \sqrt{r})]$ is $\frac{1}{2} E [Δ f (z + \sqrt{r} G)]$ by step 1.3.

F4F8step 1.3

3.1

Let $f$ and $Q$ be as in step 2.2. Continuity of $Δ f$ and bounded convergence imply that $r \mapsto Q_{r} Δ f (z)$ is continuous, including at zero. For $0 < δ < h$ , the fundamental theorem of calculus applied to the continuous integrand $r \mapsto \frac{1}{2} Q_{r} Δ f (z)$ on $[δ, h]$ gives $\frac{1}{2} \int_{δ}^{h} Q_{r} Δ f (z) d r = Q_{h} f (z) - Q_{δ} f (z)$ by step 2.2; [F8] identifies this compact calculus integral with the Lebesgue integral. Letting $δ ↓ 0$ , dominated convergence gives $Q_{δ} f (z) \to f (z)$ because $f$ is continuous and bounded, and the integrals converge by monotone convergence on the nonnegative and negative parts; hence $Q_{h} f (z) - f (z) = \frac{1}{2} \int_{0}^{h} Q_{r} Δ f (z) d r$ for every $h > 0$ .

F4F8step 2.2

4.1

Define $M_{t} : = Φ (Z_{t}) - Φ (Z_{0}) - \frac{1}{2} \int_{0}^{t} Δ Φ (Z_{r}) d r$ . The map $(r, ω) \mapsto Z_{r} (ω)$ restricted to $[0, t]$ is $B ([0, t]) \otimes G_{t}$ -measurable: finite deterministic grid approximations to the continuous paths, using only coordinates at times at most $t$ , converge pointwise. Parameter integration therefore makes the drift integral $G_{t}$ -measurable. Thus $M$ is adapted, continuous on every path and integrable on each finite horizon, with $∣ M_{t} ∣ \leq 2 ∥ Φ ∥_{\infty} + (t / 2) ∥ Δ Φ ∥_{\infty}$ . For $s \leq t$ and $A \in G_{s}$ , Fubini on the bounded finite-time integrands and step 1.4 give $\int_{A} \int_{s}^{t} Δ Φ (Z_{r}) d r d P_{x} = \int_{A} \int_{0}^{t - s} Q_{u} Δ Φ (Z_{s}) d u d P_{x} .$ By step 3.1 the inner integral is $2 (Q_{t - s} Φ (Z_{s}) - Φ (Z_{s}))$ , while step 1.4 gives $\int_{A} Φ (Z_{t}) d P_{x} = \int_{A} Q_{t - s} Φ (Z_{s}) d P_{x}$ . Thus $\int_{A} (M_{t} - M_{s}) d P_{x} = 0$ , and $M$ is a martingale.

F4F6step 1.2step 3.1step 1.4

5.1

Fix $n \geq 1$ and let $H_{n} : = H \land n$ . For each $m \geq 1$ put $δ : = 2^{- m}$ , define the discrete filtration $D_{k} : = G_{k δ}$ and the discrete martingale $Y_{k} : = M_{k δ}$ , which satisfies $E [Y_{k + 1} ∣ D_{k}] = Y_{k}$ by [F5] and step 4.1. The integer-valued ceiling $ρ_{m} : = ⌈ 2^{m} H_{n} ⌉$ is a stopping time for $(D_{k})$ , since ${ρ_{m} \leq k} = {H_{n} \leq k δ} \in G_{k δ} = D_{k}$ by step 1.1, and it is bounded by $⌈ 2^{m} n ⌉$ . Applying [F5] with $σ = 0$ and $τ = ρ_{m}$ gives $E [M_{ρ_{m} δ}] = 0$ .

F5step 1.1step 4.1

6.1

Expanding step 5.1 and letting $m \to \infty$ , $ρ_{m} δ ↓ H_{n}$ and $Z_{ρ_{m} δ} \to Z_{H_{n}}$ by continuity. Dominated convergence on $[0, n + 1]$ gives $E_{x} [Φ (Z_{H_{n}})] = Φ (x) + \frac{1}{2} E_{x} \int_{0}^{H_{n}} Δ Φ (Z_{r}) d r .$ Since $∣ Z_{r} - y ∣ \in (ε, R)$ for $r < H$ , the integral vanishes, and $E_{x} [Φ (Z_{H_{n}})] = \log ∣ x - y ∣$ .

F4step 1.1step 5.1

7.1

Define $Z_{H}$ by literal evaluation when $H < \infty$ and as $y$ otherwise. This is measurable by finite-grid approximation to $Z_{H \land n}$ and passage to the limit on ${H < \infty}$ . Letting $n \to \infty$ , continuity and bounded convergence give $E_{x} [Φ (Z_{H})] = \log ∣ x - y ∣$ . Moreover $H = S_{ε} \land T_{R}$ and $∣ Z_{H} - y ∣ \in {ε, R}$ almost surely, so $Φ (Z_{H}) = \log ε$ on ${S_{ε} < T_{R}}$ and $\log R$ on ${T_{R} < S_{ε}}$ . The tie event can include paths with both times infinite, but it has $P_{x}$ -probability zero because $H < \infty$ almost surely; a finite tie is impossible when $ε < R$ .

step 1.1step 2.1step 6.1

8.1

Therefore $\log ∣ x - y ∣ = \log ε P_{x} (S_{ε} < T_{R}) + \log R P_{x} (T_{R} < S_{ε})$ , and the two probabilities sum to one by step 7.1. Solving gives the stated formula.

step 1.2step 7.1

9.1

The hypotheses are exactly those used: $0 < ε < ∣ x - y ∣ < R$ makes $\log$ finite and the end annulus nondegenerate, the case $x = y$ is excluded, the degenerate case $ε = R$ is excluded because the formula's denominator vanishes there, and the truncated times $H_{n}$ are bounded so that the discrete optional sampling theorem applies. AC covers [F7] and the Countable Choice bridge in [F8], and no countable or dependent choice beyond AC is spent: the integration by parts, the fundamental theorem of calculus and the optional sampling theorem used here are the compact and discrete statements cited in [F4]-[F5].

F4F5F7givenstep 8.1∎

Source notes

Sousi, Section 6.7 and printed pp. 63--64, computes the annular exit probability from $\log ∣ z - y ∣$ . Durrett, Theorem 9.1.1, Lemma 9.1.3 and formula (9.1.2), gives the same harmonic-martingale calculation and planar formula. The proof above makes the harmonic martingale rigorous with an explicitly spliced bounded $C^{2}$ extension, a Gaussian integration-by-parts identity and discrete optional sampling at dyadic ceilings.

RemarkRemark: AI-adaptedProof: Not applicableaudited 2026-09-22Open item page →

Raw versus usual filtrations in the strong Markov theorem

Remark

The strong Markov theorem Strong Markov property of Brownian motion is stated for the usual augmentation $(F_{t})$ , not for the raw natural filtration $(F_{t}^{0})$ Natural and usual augmented Brownian filtrations. Four distinctions matter and are fixed by that choice.

The theorem names the filtration it uses. Its hypothesis is that $τ$ is a stopping time for $(F_{t})$ with $τ < \infty$ almost surely; the conclusion is an identity of conditional expectations given $F_{τ}$ . Neither the raw filtration nor the completed raw filtration is substituted for the usual one in the statement.
What the dyadic proof actually uses. The ceiling times $τ_{n} = 2^{- n} ⌈ 2^{n} τ ⌉$ are stopping times of the same filtration and satisfy $F_{τ} \subseteq F_{τ_{n}}$ ; the countably valued case applies the deterministic future-path theorem Future-path Markov property at the countably many values of $τ_{n}$ ; and the passage to general $τ$ uses path continuity and dominated convergence. Completion enters through the null event ${τ = \infty}$ , on which $B_{τ}$ is defined by a convention, and through the identification of conditional laws up to null sets.
Completion is not independence from arbitrary future information. The theorem asserts that the increment process $(B_{τ + t} - B_{τ})_{t \geq 0}$ is independent of $F_{τ}$ and that the conditional law of the shifted future path is Wiener measure translated by $B_{τ}$ . For $τ$ equal to a deterministic time $s > 0$ this is not the claim that the future path $(B_{s + t})_{t \geq 0}$ is independent of $F_{s}$ : its conditional law depends on the state $B_{s}$ through the translation, and only the increment process is independent of the past. No completion of the filtration removes that dependence, and none of the items on this page asserts it.
The stopping-time hypothesis is not decorative. For a random time that is not a stopping time the conclusion can fail outright; the companion example cex-strong-markov-fails-at-a-nonstopping-random-time on the companion examples page exhibits the last zero before a fixed time, where the post-time future has no zero in a right-neighbourhood and therefore cannot have the Wiener law.

The strict and non-strict forms of the stopping tests agree for the usual augmentation because it is right-continuous, while the raw statements on this page use the non-strict test directly; the ceiling identity ${τ_{n} \leq t} = {τ \leq 2^{- n} ⌊ 2^{n} t ⌋}$ is a non-strict test and needs no right-continuity. AC is declared because the conditional-expectation interface and the ambient Brownian construction assume it.

Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.

Source notes

Sousi, Sections 6.3-6.5, distinguishes the natural filtration from its right-continuous completion and states the strong Markov property for the latter; Durrett, Section 7.3, works throughout with the completed filtration. The counterexample on the companion page is oriented as a boundary for the stopping-time hypothesis, not used as a supplier anywhere in this pair.

5 · Examples, counterexamples and false statements

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