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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.

Depends on

Used by

One-dimensional Brownian motion is recurrent Corollary
Almost-sure finiteness does not imply integrability Counterexample
A Brownian hitting time has infinite mean Example
Exit side from an interval Example
Planar coordinate hitting does not imply point hitting Example
Planar Brownian annular exit probability Lemma
Two-sided Brownian exit probability Theorem

Dependency tree · two levels

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Sources

Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.4 (standard reference, not scraped)
Perla Sousi, Advanced Probability, Section 6.7 (standard reference, not scraped)