Brownian Path Properties — Examples

For each $k$ the increment ${\mathrm{\Delta }}_k$ has the law $N(0,h)$ by [F2], so $E{\mathrm{\Delta }}_k^2=h$.

By [F1] the terminal dyadic sum is ${\sum }_{k=1}^{2^n}{\mathrm{\Delta }}_k^2$, and linearity of expectation with [step 1.1] gives $E{\sum }_k{\mathrm{\Delta }}_k^2={\sum }_{k}h=2^n\cdot T\mathrm{/}{2}^n=T$.

The cases are covered: $T>0$ and $n\ge 1$ give $h>0$ and $2^n$ summands, including the degenerate case $n=1$; the mean computation uses only the marginal law of each increment, not independence or any joint distribution; and AC enters only through [F3].

For each $k$, [F1] gives $\mathrm{Var}({\mathrm{\Delta }}_k^2)=E{\mathrm{\Delta }}_k^4-(E{\mathrm{\Delta }}_k^2{)}^2=3h^2-h^2=2h^2$.

The variables ${\mathrm{\Delta }}_k^2$ are functions of increments over disjoint intervals, hence independent by [F1], so the variance of the sum is the sum of the variances: $\mathrm{Var}(Q_n)={\sum }_{k=1}^{2^n}2h^2=2^n\cdot 2(T\mathrm{/}{2}^n{)}^2=2T^2\mathrm{/}{2}^n$.

Since $E{Q}_n=T$ by [F2], $E(Q_n-T{)}^2=\mathrm{Var}(Q_n)=2T^2\mathrm{/}{2}^n\to 0$, which is the $L^2$ convergence $Q_n\to T$.

The cases are covered: the independence of the squared increments is the only place where the joint law is used; the value $n=1$ is included and gives $\mathrm{Var}(Q_1)=T^2$; the limit is taken as $n\to \mathrm{\infty }$ with $T$ fixed and positive; and AC enters only through [F3].

Intersecting the probability-one events supplied by [F1] for the positive integer horizons with the single probability-one event of [F2] gives a probability-one event on which nullity holds at every positive integer horizon and uncountability holds at every positive horizon. For arbitrary $T>0$, choose an integer $N\ge T$; then $\lambda (Z\cap [0,T])\le \lambda (Z\cap [0,N])=0$, while uncountability of $Z\cap [0,T]$ follows directly from [F2].

On that event the set $Z\cap [0,T]$ is closed by [F3] and compact, uncountable by [step 1.1], and null by [step 1.1]; the two properties are not in tension because uncountability imposes no lower bound on Lebesgue measure, as the Cantor set shows in the deterministic setting.

The cases are covered: the horizon is positive and finite; simultaneous nullity for all horizons is obtained from the countable integer exhaustion, whereas simultaneous uncountability for all positive horizons is exactly [F2]; the point $0$ belongs to $Z$ and to every truncated set but is a singleton of measure zero; and AC enters only through [F4].

If $x$ is constant all its variation sums are zero, and if $p=q$ the comparison is equality. Otherwise fix a nonconstant continuous $x$ on $[a,b]$ with finite oscillation $M:={\max }_{[a,b]}x-{\min }_{[a,b]}x$, and let $1\le p<q$; for every partition, ${\sum }_i\mathrm{\mid }{x}_{t_{i+1}}-x_{t_i}{\mathrm{\mid }}^q\le M^{q-p}{\sum }_i\mathrm{\mid }{x}_{t_{i+1}}-x_{t_i}{\mathrm{\mid }}^p$, so taking suprema gives $V_q(x;[a,b])\le M^{q-p}{V}_p(x;[a,b])$; consequently $V_p<\mathrm{\infty }$ implies $V_q<\mathrm{\infty }$ and $V_q=\mathrm{\infty }$ implies $V_p=\mathrm{\infty }$.

Suppose that for some continuous $x$ on $[a,b]$ and some $1\le p<2$ one had $V_p(x;[a,b])<\mathrm{\infty }$; then for the dyadic partition with mesh $h=(b-a)\mathrm{/}{2}^n$ one has ${\sum }_k({\mathrm{\Delta }}_{k}x{)}^2={\sum }_k\mathrm{\mid }{\mathrm{\Delta }}_{k}x{\mathrm{\mid }}^{2-p}\mathrm{\mid }{\mathrm{\Delta }}_{k}x{\mathrm{\mid }}^p\le ({\max }_k\mathrm{\mid }{\mathrm{\Delta }}_{k}x\mathrm{\mid }{)}^{2-p}{V}_p(x;[a,b])$, and the maximum increment tends to $0$ by uniform continuity of the continuous $x$ on the compact $[a,b]$, so those quadratic sums would tend to $0$.

Suppose now that a continuous $x$ on $[a,b]$ satisfies $V_2(x;[a,b])<\mathrm{\infty }$, and put $v(a)=0$ and $v(t):=V_2(x;[a,t])$ for $a<t\le b$; then $v$ is nondecreasing and finite-valued, and for $t<b$, $h>0$ with $t+h\le b$ one has $\mathrm{\mid }{x}_{t+h}-x_t{\mathrm{\mid }}^2\le v(t+h)-v(t)$ by adding the single point $t+h$ to partitions of $[a,t]$ and taking suprema.

For each fixed deterministic $t\ge 0$ the process $Y_h=B_{t+h}-B_t$, $h\ge 0$, starts at zero and has continuous paths. Its increments on disjoint $h$-intervals are increments of $B$ on disjoint time-translated intervals, so they have the independent centered normal laws with the required lengths. Thus $Y$ is standard Brownian motion. Applying [F2] to $Y$ with $t=a$ and $T=b-a$ identifies its dyadic sums term by term with those on $[a,b]$ and proves their almost-sure limit $b-a$ for each fixed interval. Applying [F3] to $Y$ at each fixed $t$ gives ${\mathrm{lim\hspace{0.17em}sup}}_{h\downarrow 0}\mathrm{\mid }{B}_{t+h}-B_t{\mathrm{\mid }}^2\mathrm{/}h=\mathrm{\infty }$ almost surely: along a sequence where the LIL ratio is at least $1\mathrm{/}2$, that quotient is at least $\frac{1}{2}\log \log (1\mathrm{/}h)\to \mathrm{\infty }$.

Let $p>2$ be rational and choose a rational $\gamma$ with $1\mathrm{/}p<\gamma <1\mathrm{/}2$; by [F1] there is, on a probability-one event, a finite $K$ with $\mathrm{\mid }{B}_t-B_s\mathrm{\mid }\le K\mathrm{\mid }t-s{\mathrm{\mid }}^{\gamma }$ on $[0,b]$; then for every partition of $[a,b]$, ${\sum }_i\mathrm{\mid }{B}_{t_{i+1}}-B_{t_i}{\mathrm{\mid }}^p\le K^p{\sum }_i(t_{i+1}-t_i{)}^{\gamma p}\le K^p(b-a{)}^{\gamma p}$ because $\gamma p>1$ and ${\sum }_i{h}_i^{\gamma p}\le ({\sum }_i{h}_i{)}^{\gamma p}$; hence $V_p(B;[a,b])\le K^p(b-a{)}^{\gamma p}<\mathrm{\infty }$ almost surely.

But by step 1.4 the dyadic quadratic sums of the Brownian path converge to $b-a>0$ almost surely, so the hypothesis of [step 1.2] fails for $B$ on every rational interval $[a,b]$ and rational $p\in [1,2)$ almost surely; intersecting countably many events and using [step 1.1] to pass to every real $p\in [1,2)$ gives $V_p(B;[a,b])=\mathrm{\infty }$ almost surely for every $p\in [1,2)$ and every nondegenerate compact interval.

By [F4] the function $v$ is differentiable at Lebesgue-almost every $t\in (a,b)$ with finite derivative, so at those $t$ one has ${\mathrm{lim\hspace{0.17em}sup}}_{h\downarrow 0}\mathrm{\mid }{x}_{t+h}-x_t{\mathrm{\mid }}^2\mathrm{/}h\le v^{\mathrm{\prime }}(t)<\mathrm{\infty }$.

Let $N:=\{t\in [a,b]:{\mathrm{lim\hspace{0.17em}sup}}_{h\downarrow 0,h\in \mathbb{Q}}\mathrm{\mid }{B}_{t+h}-B_t{\mathrm{\mid }}^2\mathrm{/}h<\mathrm{\infty }\}$; the set is product measurable: for each rational $h>0$ the map $(t,\omega )\mapsto (t+h,\omega )$ is product measurable, so joint measurability of $B$ makes each difference quotient measurable, and the limsup is the infimum over positive integers $n$ of the countable suprema over rational $0<h<1\mathrm{/}n$. Its indicator is integrable since the product space has total measure $b-a$. For every fixed deterministic $t\in [a,b)$ the shifted process $h\mapsto B_{t+h}-B_t$ is again a standard Brownian motion by [F3], so the divergence recorded in [step 1.4] holds for that $t$: along a sequence $h_k\downarrow 0$ the squared difference quotient is unbounded. Since the path $h\mapsto B_{t+h}$ is continuous, the function $h\mapsto \mathrm{\mid }{B}_{t+h}-B_t{\mathrm{\mid }}^2\mathrm{/}h$ is finite-valued and continuous on $(0,\mathrm{\infty })$, so for every $\delta >0$ its supremum over the dense subset $\mathbb{Q}\cap (0,\delta )$ equals its supremum over all of $(0,\delta )$, the limsup along rational $h\downarrow 0$ coincides with the limsup along real $h\downarrow 0$, and the latter is $+\mathrm{\infty }$ almost surely; hence $P(t\in N)=0$ for every $t\in [a,b)$ and not merely for the rational ones. Therefore [F5] gives $E\lambda (N)={\int }_a^{b}P(t\in N)dt=0$ and hence $\lambda (N)=0$ almost surely.

Intersecting the events of [step 2.1] over the countably many rational $p>2$ and rational pairs $a<b$, and using [step 1.1] to pass from a rational $p$ to every real $q>2$ with $p\le q$, we obtain: almost surely $V_q(B;[a,b])<\mathrm{\infty }$ for every $q>2$ and every compact interval with rational endpoints, hence by containment for every nondegenerate compact interval.

First intersect the events from step 2.4 over all rational $0\le a<b$. On the resulting probability-one event the set of $t$ at which [step 2.3] would hold has measure zero, so a continuous path with $V_2(x;[a,b])<\mathrm{\infty }$ cannot be a Brownian path: almost surely $V_2(B;[a,b])=\mathrm{\infty }$ for every rational interval and hence every nondegenerate compact interval in $[0,\mathrm{\infty })$: it contains a nondegenerate rational subinterval, and any partition of the latter extends to a partition of the former by adjoining the two outer endpoints; all extra terms are nonnegative.

Combining [step 3.1], [step 2.2] and [step 3.2] on the intersection of the countably many probability-one events, almost surely $V_p(B;[a,b])<\mathrm{\infty }$ for all $p>2$ and $V_p(B;[a,b])=\mathrm{\infty }$ for all $1\le p\le 2$, simultaneously on every nondegenerate compact interval; the exponent $p=1$ case is the infinite total variation of the path, and the value $p=2$ is handled by the accumulated-variation argument rather than by the dyadic sums, which are finite.

The boundary cases are covered: $p\ge 1$ is required by the statement, so no fractional exponents below one occur; the interval is nondegenerate and compact, and rational endpoints suffice by [F6]; the two-variation $V_2$ is a supremum over all partitions and is deliberately distinguished from the dyadic quadratic sums, which converge almost surely to $b-a$ for each fixed interval by step 1.4; the monotonicity of [step 1.1] transfers between exponents using the finite oscillation of a continuous function on a compact interval; AC${}_{\omega }$ is declared exactly at the monotone-differentiability interface [F4], and AC is the ambient assumption of [F6].

On the probability-one event of [F1], combining the two limit statements gives ${\mathrm{lim\hspace{0.17em}sup}}_{t\downarrow 0}\mathrm{\mid }{B}_t\mathrm{\mid }\mathrm{/}\sqrt{2t\log \log (1\mathrm{/}t)}=1$, so there are $t_n\downarrow 0$ with $\sqrt{2\log \log (1\mathrm{/}{t}_n)}\mathrm{\mid }{B}_{t_n}\mathrm{\mid }\mathrm{/}\sqrt{2t_n\log \log (1\mathrm{/}{t}_n)}\to \mathrm{\infty }$.

For such a sequence $\mathrm{\mid }{B}_{t_n}\mathrm{\mid }\mathrm{/}\sqrt{t_n}=\sqrt{2\log \log (1\mathrm{/}{t}_n)}\cdot \frac{\mathrm{\mid }{B}_{t_n}\mathrm{\mid }}{\sqrt{2t_n\log \log (1\mathrm{/}{t}_n)}}\to \mathrm{\infty }$; hence for every finite constant $c$ there exist $t\in (0,\delta )$ with $\mathrm{\mid }{B}_t\mathrm{\mid }>c\sqrt{t}$, whatever $\delta >0$ is prescribed, which is exactly the failure of the displayed bound for a finite random $C$ and random $\delta >0$.

The cases are covered: only $t>0$ is quantified, so the normalizer $\log \log (1\mathrm{/}t)$ is defined and positive for small $t$; the constant $C$ is allowed to be random and finite, and the argument produces, on the given outcome, arbitrarily small times violating any fixed finite value; and AC enters only through [F2].

For $m\ge 1$ put $I_m:=[2^{-(m+1)},2^{-m}]$, $n_m:=2m^4$, ${\delta }_m:=2^{-(m+2)}\mathrm{/}{m}^4=2^{-(m+1)}\mathrm{/}{n}_m$, and let $s_{m,j}:=2^{-(m+1)}+j{\delta }_m$ for $0\le j\le n_m$; define $x(s_{m,j}):=1\mathrm{/}m$ for odd $j$ and $x(s_{m,j}):=0$ for even $j$, interpolate $x$ linearly between consecutive vertices of each block, and set $x:=0$ on $\{0\}\cup [1\mathrm{/}2,1]$.

The function $x$ is well defined and continuous: on each block it is piecewise linear hence continuous, the last vertex of $I_m$ has even index and value $0$, matching the value $0$ at the shared endpoints of consecutive blocks and on $[1\mathrm{/}2,1]$, and for $t\in (0,2^{-m}]$ one has $\mathrm{\mid }x(t)\mathrm{\mid }\le 1\mathrm{/}m$, so $x(t)\to 0=x(0)$ as $t\downarrow 0$.

For $n\ge 1$ let ${\pi }_n$ be the partition of $[0,1]$ whose point set is the dyadic grid $\{k{2}^{-n}:0\le k\le 2^n\}$ together with every vertex $s_{m,j}$ with $1\le m\le n$; each ${\pi }_n$ is a finite partition in the sense of Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, the sequence is refining, and $\mathrm{mesh}({\pi }_n)\le 2^{-n}$.

No dyadic point of level $n$ lies in the interior of $I_n$: such a point would be $k{2}^{-n}$ with $1\mathrm{/}2<k<1$, and no integer satisfies this; hence the points of ${\pi }_n$ inside $I_n$ are exactly the vertices $s_{n,0}<\cdots <s_{n,n_n}$, and consecutive points of ${\pi }_n$ inside that block are consecutive vertices.

The quadratic sum along ${\pi }_n$ therefore contains the $n_n$ vertex increments of the block $I_n$, each of absolute value $1\mathrm{/}n$, so it is at least $n_n\cdot (1\mathrm{/}n{)}^2=2n^4\mathrm{/}{n}^2=2n^2$; since $2n^2\to \mathrm{\infty }$, the quadratic sums along the refining sequence $({\pi }_n)$ diverge to $+\mathrm{\infty }$ for this continuous function.

Consequently continuity alone does not force finite quadratic variation along a prescribed refining sequence with vanishing mesh: the witness is the explicit sawtooth function above, whose block $I_n$ alone contributes $2n^2$ to the $n$-th quadratic sum; the example also shows that the mesh condition of Quadratic variation along a partition sequence is not sufficient by itself, and the construction uses no choice principle.

The probability-one event in [F1] is nonempty; fix an outcome $\omega$ in that event, so the same chosen path is continuous and has both stated variation properties.

For the path $t\mapsto B_t(\omega )$ on $[0,1]$ the dyadic quadratic sums are finite for each mesh and converge uniformly to elapsed time, whereas the supremum of the absolute-increment sums over partitions of $[0,1]$ is $+\mathrm{\infty }$; both assertions refer to the same fixed continuous path.

Hence the refuted implication fails: the witness is continuous on $[0,1]$, has finite quadratic variation there in the stated dyadic sense, and nevertheless is not of bounded variation there. AC is the standing hypothesis required by the Brownian corollary [F1], as recorded in [F2]; after its nonempty probability-one event is supplied, fixing one witness makes no additional choice-family construction.

The profile is continuous and differentiable at the origin: $G(s)=s^{2}T(s)$ is a product of continuous functions, and with $M:={\sup }_{[0,1]}\mathrm{\mid }T\mathrm{\mid }<\mathrm{\infty }$, finite because the continuous $T$ is bounded on the compact interval, one has $\mathrm{\mid }G(s)\mathrm{\mid }\le M{s}^2$ for every $s\in [0,1]$, so $G(s)\mathrm{/}s\to 0$ and $G^{\mathrm{\prime }}(0)=0$.

The profile is nowhere else differentiable: if $G$ were differentiable at some $s\in (0,1]$, then $T=G\mathrm{/}{s}^2$ would be differentiable at $s$ as a quotient with nonvanishing denominator, contradicting the nowhere differentiability of the Takagi function on $[0,1]$.

Every sample path $t\mapsto X_t(\omega )=G(\mathrm{\mid }t-\omega \mathrm{\mid })$ is continuous, being a composition of the continuous maps $t\mapsto \mathrm{\mid }t-\omega \mathrm{\mid }$ and $G$; for the same reason the map $(t,\omega )\mapsto X_t(\omega )$ is jointly measurable.

At $t=\omega$ the path is differentiable with derivative $0$: for $h\mathrm{\ne }0$ the difference quotient is $(G(\mathrm{\mid }h\mathrm{\mid })-G(0))\mathrm{/}h=\pm G(\mathrm{\mid }h\mathrm{\mid })\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$, whose limit as $h\to 0$ is $\pm G^{\mathrm{\prime }}(0)=0$ by [step 1.1].

At every $t\in (0,1)$ with $t\mathrm{\ne }\omega$ the path is not differentiable at $t$: on a neighbourhood of such a $t$ that lies inside $(0,1)$ the path is the composition of the affine map $u\mapsto \pm (u-\omega )$ of nonzero slope with the restriction of $G$, so differentiability of the path at $t$ would make $G=X\circ A^{-1}$ differentiable at $\mathrm{\mid }t-\omega \mathrm{\mid }$, which lies in $(0,1)$ because $t\in (0,1)$ and $\omega \in [0,1]$, and [step 2.1] excludes exactly that.

For a fixed deterministic $t\in (0,1)$ the path is differentiable at $t$ exactly when $\omega =t$, by [step 2.3] and [step 3.1]; since the singleton $\{t\}$ is Lebesgue-null, the path is almost surely not differentiable at $t$, and this holds for every $t$ of the uncountable family $(0,1)$.

Yet almost surely the path is differentiable somewhere, namely at $t=\omega \in (0,1)$, by [step 2.3]; hence the pathwise event "the path is nowhere differentiable on $(0,1)$" has probability $0$. The fixed-time assertions of [step 4.1] therefore do not upgrade to the pathwise statement: the quantifier over the uncountable family of times cannot be moved inside the almost-sure statement, which is the defect being exhibited.

The boundary and degenerate cases are covered: the fixed-time family is the open interval $(0,1)$, so that at each of its times the two-sided notion of differentiability applies and the endpoint behaviour of the profile is never needed; the case $t=\omega$ is the differentiability point of the path and contributes the null singleton to the fixed-time computation of [step 4.1]; the outcomes $\omega \in \{0,1\}$ form a null edge case and are not needed, since [step 5.1] only requires the event of positive probability on which the path is differentiable at an interior time; the profile is not constant, so the degenerate case in which every time were a differentiability point does not arise; and the construction selects nothing, the measure space and the profile being explicit.