Brownian Path Properties

Fix rationals $0\le a<b$ and an integer $C\ge 1$, and put $E_{a,b,C}:={\bigcap }_{r,q\in \mathbb{Q}\cap [a,b]}\{\mathrm{\mid }{B}_r-B_q\mathrm{\mid }\le C\mathrm{\mid }r-q{\mathrm{\mid }}^{1\mathrm{/}2}\}\mathrm{.}$ This is measurable because it is a countable intersection of coordinate events. For every integer $n\ge 1$, with $h=(b-a)\mathrm{/}n$ and grid points $t_k=a+kh$ (all rational), the event $E_{a,b,C}$ is contained in $A_n:=\{\omega :\mathrm{\mid }{B}_{t_k}(\omega )-B_{t_{k-1}}(\omega )\mathrm{\mid }\le C\sqrt{h}$ for $k=1,\dots ,n\}$, because consecutive grid points are rational pairs in $[a,b]$ at distance $h$.

Insert the endpoint 0 before a when a>0; [F1] then applies to the increasing grid starting at zero, and its subfamily of increments on [a,b] is independent. By [F1] and [F2] the increments $B_{t_k}-B_{t_{k-1}}$, $k=1,\dots ,n$, are independent with the law of $\sqrt{h}Z$, so each satisfies $P(\mathrm{\mid }{B}_{t_k}-B_{t_{k-1}}\mathrm{\mid }\le C\sqrt{h})=P(\mathrm{\mid }Z\mathrm{\mid }\le C)=p_C<1$, and independence gives $P(A_n)=p_C^n$; hence $P(E_{a,b,C})\le p_C^n$ for every $n\ge 1$ and therefore $P(E_{a,b,C})=0$. For completeness, the standard normal probability of [C+1,C+2] is at least $e^{-(C+2{)}^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }>0$, proving p_C<1 for the positive integers C used here.

The family of ordered pairs of rationals and of integers is countable, so [step 2.1], countable subadditivity [F5] and [F4] give $P({\bigcup }_{0\le a<b,a,b\in \mathbb{Q}}{\bigcup }_{C\ge 1}{E}_{a,b,C})=0$.

On the complement of that null event there is no nondegenerate interval $I$ with a finite one-half Hölder constant: if $I$ were such an interval with any finite real constant $C$, then by [F3] we could choose rationals $0\le a<b$ with $[a,b]\subseteq I$, and with $C^{\mathrm{\prime }}:=\lceil \max (C,1)\rceil \in \mathbb{N}$ the bound would in particular hold for all rational $s,t\in [a,b]$, that is, $E_{a,b,C^{\mathrm{\prime }}}$ would occur.

The intended cases are covered: the interval is required to be nondegenerate, so the empty and singleton interval cases are excluded; the value $n=1$ in [step 2.1] is the degenerate single-increment case of the estimate and already gives $P(A_1)=p_C<1$; the union over integers $C\ge 1$ covers every finite real constant up to rounding up; the estimates in [step 2.1] hold for every positive integer n and imply nullness without requiring the mesh events to be nested; and AC is used only through [F4] via [F1] and [F2].

Suppose the continuous path $f:=B(\omega )$ has a finite two-sided derivative at some $s\in (a,b)$, or a finite right derivative at $s=a$, or a finite left derivative at $s=b$, with absolute value at most $C$; use both sides for an interior point, the right side at a and the left side at b, applying [F2] to obtain $\delta >0$ such that $\mathrm{\mid }f(t)-f(s)\mathrm{\mid }\le (C+1)\mathrm{\mid }t-s\mathrm{\mid }$ for every $t\in [a,b]$ on the permitted side or sides of $s$ with $0<\mathrm{\mid }t-s\mathrm{\mid }<\delta$.

Fix $n\ge 6$ with $(b-a)\mathrm{/}n<\delta \mathrm{/}6$, write $h:=(b-a)\mathrm{/}n$ and $t_k:=a+kh$, and put $k_0:=\max \{k\in \{0,\dots ,n-1\}:t_k\le s\}$. Then $t_{k_0}\le s\le t_{k_0+1}$, including $k_0=n-1$ when $s=b$. If $k_0+5\le n$ take the block of five increments beginning at $t_{k_0}$, and otherwise take the block of five increments ending at $t_n$. In either case all endpoints lie in $[a,b]$, on the side of $s$ allowed in step 1.1 for the endpoint cases, and within distance $6h<\delta$ of $s$, so each increment has absolute value at most $2(C+1)\cdot 6h=12(C+1)h=:D\mathrm{/}n$ with $D:=12(C+1)(b-a)$.

For n=0,1,2,3,4,5 set G_n to be the empty event, without defining h or a mesh for those indices. For integers n>=6 define $G_n$ to be the event that some block of five consecutive increments $B_{t_{k+i}}-B_{t_{k+i-1}}$ ($k=0,\dots ,n-5$, $i=1,\dots ,5$) has all five absolute values at most $D\mathrm{/}n$, where $h=(b-a)\mathrm{/}n$. Every G_n is a finite union of finite intersections of measurable coordinate events. Insert 0 before a if a>0 and use the subfamily of grid increments in [a,b]; the increments of one block are independent with laws $N(0,h)$ by [F1], so by [F3] and independence the probability for a fixed block is at most $y_n^5$, where $y_n:=D\mathrm{/}\sqrt{(b-a)n}=12(C+1)\sqrt{(b-a)\mathrm{/}n}\le 1$ for large $n$. Hence $P(G_n)\le n{y}_n^5={12}^5(C+1{)}^5(b-a{)}^{5\mathrm{/}2}{n}^{-3\mathrm{/}2}$, and ${\sum }_{n}P(G_n)<\mathrm{\infty }$: the finitely many remaining initial terms are at most one each, and ${\sum }_{n=2^j}^{2^{j+1}-1}{n}^{-3\mathrm{/}2}\le 2^{-j\mathrm{/}2}$ bounds the tail by a geometric series.

By [F4] and step 3.1, almost surely $G_n$ fails for all sufficiently large $n$; by step 2.1 this means that almost surely the path has no finite derivative with absolute value at most $C$ at any point of $[a,b]$ (two-sided on $(a,b)$, right at $a$, left at $b$).

Intersect the common continuity event from [F1] with the complements of all the measurable limsup events of [F4]. Taking the union of those null events over the countably many rational pairs $0\le a<b$ and over integers $C\ge 1$, and using [F5] to place every $s>0$ in the interior of such an interval (while $s=0$ is the left endpoint of one), we obtain: almost surely no time $s\ge 0$ has a finite two-sided derivative (for $s>0$) or finite right derivative (for $s=0$).

The boundary cases are covered by the block choices of step 2.1: $s=0$ uses the right-handed block beginning at $a$, $s=b$ the left-handed block ending at $b$, and interior times either the forward or the backward block, all of which stay inside $[a,b]$; the cases n<6 are defined to be empty events in step 3.1, so the sequence is indexed by all natural numbers and no division by zero is performed; rounding the derivative bound up to an integer $C$ loses nothing, and the finite-difference ratio of [F2] is the definition-level form of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set; AC is inherited through [F6] from the Brownian and normal-law interfaces.

For $Z\sim N(0,1)$ one has $E\mathrm{\mid }Z\mathrm{\mid }=\sqrt{2\mathrm{/}\pi }$ and $\mathrm{Var}(\mathrm{\mid }Z\mathrm{\mid })=1-2\mathrm{/}\pi$: by [F2] and [F3], $E\mathrm{\mid }Z\mathrm{\mid }=2{\int }_0^{\mathrm{\infty }}x\phi (x)dx=2(2\pi {)}^{-1\mathrm{/}2}{\lim }_{R\to \mathrm{\infty }}(1-e^{-R^2\mathrm{/}2})=(2\mathrm{/}\pi {)}^{1\mathrm{/}2}$, while $E\mathrm{\mid }Z{\mathrm{\mid }}^2=E{Z}^2=1$ by [F9] with $m=1$, so [F10] gives variance $1-(2\mathrm{/}\pi )$. The compact substitution integrals in [F3] are transferred to Lebesgue integrals before the nonnegative monotone limit; expectation and density identities in [F3] justify the displayed Gaussian integral.

Fix $n\ge 1$, put $h=(b-a)\mathrm{/}{2}^n$ and $S_n:={\sum }_{k=1}^{2^n}\mathrm{\mid }{B}_{a+kh}-B_{a+(k-1)h}\mathrm{\mid }$; by [F1] and [F2] the $2^n$ increments are independent with the law of $\sqrt{h}Z$, so $E{S}_n=2^n\sqrt{2h\mathrm{/}\pi }=\sqrt{2(b-a)\mathrm{/}\pi }\cdot 2^{n\mathrm{/}2}\to \mathrm{\infty }$ and $\mathrm{Var}(S_n)=2^{n}h(1-2\mathrm{/}\pi )=(b-a)(1-2\mathrm{/}\pi )$ is constant in $n$. Indeed every absolute increment is square-integrable by step 1.1; [F10] applied to the absolute-value Borel functions on any two distinct increments gives zero covariance, and finite bilinearity gives the variance sum. If $a>0$, the independent increment list in [F1] is obtained by including $0,a$ in the time grid and discarding its first increment; for $a=0$ no extra interval is needed.

By [F5] applied to $S_n$ with $\lambda =\frac{1}{2}E{S}_n$, $P(S_n\le \frac{1}{2}E{S}_n)\le 4\mathrm{Var}(S_n)\mathrm{/}(E{S}_n{)}^2=\frac{4(b-a)(1-2\mathrm{/}\pi )}{\frac{2(b-a)}{\pi }{2}^n}=\frac{2\pi (1-2\mathrm{/}\pi )}{2^n}$, which is summable in $n$.

By [F6] and [step 2.1], almost surely $S_n>\frac{1}{2}E{S}_n$ for all sufficiently large $n$, and hence $S_n\to \mathrm{\infty }$; since each $S_n=V(B,P_n)$ is the variation of the path over the dyadic partition $P_n$ of $[a,b]$, the variation sums over partitions of $[a,b]$ are almost surely unbounded above.

The argument of steps 1.1-4.1 depends on $0\le a<b$ only through the single number $b-a$, so it applies verbatim to every ordered pair of nonnegative rationals $a<b$; by [F7] the pairs form a countable family. For each pair use the measurable event ${\bigcup }_N{\bigcap }_{n\ge N}\{S_n>\frac{1}{2}E{S}_n\}$ of probability one from step 4.1; intersecting these explicit events gives a measurable probability-one event on which the variation sums of $B$ are unbounded above on every compact interval in $[0,\mathrm{\infty })$ with rational endpoints.

On that event every nondegenerate compact interval $[c,d]\subseteq [0,\mathrm{\infty })$ has unbounded variation sums as well: choose, by [F7], nonnegative rationals $a<b$ with $c<a<b<d$, note that the dyadic partitions of $[a,b]$ extend to partitions of $[c,d]$ by adding the points $c$ and $d$, and that adding points can only increase a variation sum by the triangle inequality, so the sums over partitions of $[c,d]$ dominate the unbounded family for $[a,b]$.

The boundary cases are covered: the interval is required to be nondegenerate, so $a=b$ and the singleton convention are excluded; $2^n\ge 2$ increments are used, so the $n$-sums are genuine variation sums over partitions in the sense of [F4]; the variance bound is uniform in the mesh index for each fixed interval, while the full-measure intersection in step 5.1 is legitimate by countability, not by an interval-independent variance constant. AC includes the Countable Choice required for the compact Riemann/Lebesgue bridge and is inherited through the Brownian and normal-law interfaces in [F8].

Writing ${\mathrm{\Delta }}_k:=B_{kh}-B_{(k-1)h}$ for $k=1,\dots ,2^n$, [F1] and [F2] give $E{\mathrm{\Delta }}_k^2=h$ and $E{\mathrm{\Delta }}_k^4=3h^2$, so the centered variables $Y_k:={\mathrm{\Delta }}_k^2-h$ satisfy $E{Y}_k=0$ and $\mathrm{Var}(Y_k)=E{\mathrm{\Delta }}_k^4-(E{\mathrm{\Delta }}_k^2{)}^2=3h^2-h^2=2h^2$.

$Q_n-T={\sum }_{k=1}^{2^n}{Y}_k$ has mean $0$, and [F1] makes the $Y_k$ independent, so $\mathrm{Var}(Q_n-T)={\sum }_k\mathrm{Var}(Y_k)=2^n\cdot 2(T\mathrm{/}{2}^n{)}^2=2T^2\mathrm{/}{2}^n$; hence $E(Q_n-T{)}^2=2T^2\mathrm{/}{2}^n\to 0$ and $Q_n\to T$ in $L^2$.

For every $\epsilon >0$, [F3] gives $P(\mathrm{\mid }{Q}_n-T\mathrm{\mid }\ge \epsilon )\le \mathrm{Var}(Q_n-T)\mathrm{/}{\epsilon }^2=2T^2\mathrm{/}(2^n{\epsilon }^2)$, which is summable in $n$; applying [F4] to $G_n=\{\mathrm{\mid }{Q}_n-T\mathrm{\mid }\ge \frac{1}{m}\}$ for each $m\ge 1$ and intersecting the resulting probability-one events over $m$ shows that almost surely $Q_n\to T$.

The degeneracies are covered: $T>0$ is required, so $h>0$ and the sums are nonempty with $2^n\ge 2$ terms; the partition sequence is the one named in [F5], with mesh $T\mathrm{/}{2}^n\to 0$ and with consecutive refinements ${\pi }_n\subseteq {\pi }_{n+1}$, so no ambiguity of convention arises at $t=T$, where the step and partial-increment conventions coincide by Quadratic variation along a partition sequence; and AC enters only through [F6].

Put ${\mathrm{\Delta }}_j:=B_{t_j}-B_{t_{j-1}}$, $X_j:={\mathrm{\Delta }}_j^2-h$ and $S_j:={\sum }_{i=1}^j{X}_i$ for $j=1,\dots ,2^n$; at partition points the step convention reads $[B{]}_{t_j}^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}=S_j+t_j$, and the two conventions agree there by [F5].

By [F1] and [F2], $E{X}_j=h-h=0$ and $\mathrm{Var}(X_j)=3h^2-h^2=2h^2$; the $X_j$ are independent because they are functions of disjoint increments, so $\mathrm{Var}(S_{2^n})=2^n\cdot 2h^2=2T^2\mathrm{/}{2}^n$, and [F3] gives $P({\max }_{1\le j\le 2^n}\mathrm{\mid }{S}_j\mathrm{\mid }\ge \epsilon )\le 2T^2\mathrm{/}(2^n{\epsilon }^2)$ for every $\epsilon >0$.

The bound of [step 2.1] is summable in $n$ for each fixed $\epsilon >0$; applying [F4] to the events $\{{\max }_j\mathrm{\mid }{S}_j\mathrm{\mid }\ge 1\mathrm{/}m\}$ for $m\ge 1$ and intersecting the resulting probability-one events over $m$ yields: almost surely, for every rational $\epsilon >0$ one has ${\max }_{1\le j\le 2^n}\mathrm{\mid }{S}_j\mathrm{\mid }<\epsilon$ for all sufficiently large $n$, hence ${\max }_j\mid [B{]}_{t_j}^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}-t_j\mid \to 0$.

For $t\in [t_j,t_{j+1})$ the step convention satisfies $\mid [B{]}_t^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}-t\mid \le \mathrm{\mid }{S}_j\mathrm{\mid }+(t-t_j)\le {\max }_j\mathrm{\mid }{S}_j\mathrm{\mid }+h$, and the same bound with $j=2^n$ holds at $t=T$; since $h=T\mathrm{/}{2}^n\to 0$, [step 3.1] gives almost-sure uniform convergence to $t$ for the step convention on $[0,T]$.

For the partial-increment convention, [F5] gives $\mid [B{]}_t^{{\pi }_n,\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}-[B{]}_t^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}\mid =(B_t-B_{t_j}{)}^2\le \omega (T\mathrm{/}{2}^n{)}^2$, where $\omega (\delta ):=\sup \{\mathrm{\mid }{B}_u-B_v\mathrm{\mid }:u,v\in [0,T],\mathrm{\mid }u-v\mathrm{\mid }\le \delta \}$; by [F6] and the finite-subcover argument applied to the continuous path on the compact interval $[0,T]$, $\omega (\delta )\to 0$ as $\delta \downarrow 0$, so the two conventions have the same uniform limit $t$.

The boundary cases are covered: $T>0$ so $h>0$ and the sums have at least two terms; $t=0$ is a partition point with $[B{]}_0^{{\pi }_n}=0$ for both conventions; $t=T$ is a partition point where the conventions coincide by [F5]; the mesh tends to zero and the partitions refine, so the named sequence is a partition sequence in the sense of [F5]; and AC enters only through [F7].

By [F1] there is one probability-one event on which the variation sums are unbounded on every nondegenerate compact interval in $[0,\mathrm{\infty })$; this is already a single almost-sure statement and needs no further intersection.

For each fixed $T>0$, [F2] gives a probability-one event on which the dyadic partial quadratic-variation processes formed from the dyadic partitions of $[0,T]$ converge uniformly to $t$ on $[0,T]$. For a prescribed countable $C\subseteq (0,\mathrm{\infty })$, select such a measurable full-measure event $E_T$ for each $T\in C$ using the stated AC, and intersect with the single full-measure continuity event supplied by Brownian motion. The event $E_C={\bigcap }_{T\in C}{E}_T$ (also intersected with that continuity event) has probability one, giving simultaneous convergence for $C$. This includes finite and empty $C$ and all positive integer horizons. The partition used for each $T$ has points $kT\mathrm{/}{2}^n$; convergence is not transferred between differently scaled grids. [F3]

For a fixed horizon, or the prescribed countable set in step 1.2, intersecting the probability-one event of [step 1.1] with the corresponding event of [step 1.2] gives both assertions simultaneously; the dyadic partition sequence is named, so the second conclusion is a statement about that sequence and not about arbitrary partitions.

The degenerate cases are covered: the interval in the first assertion and the horizon $T$ in the second are required to be nondegenerate and positive respectively; the value $t=0$ is a partition point at which both quadratic sums vanish; only prescribed countable families of horizons are intersected, because the dyadic partitions supplied by [F2] depend on the horizon; and AC enters through the suppliers and the countable selection of their full-measure events in [F3].