The Brownian p-variation threshold

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].