# On the "uniform continuity" of Brownian motion under expectation

Let $$(W_t)_{t\ge 0}$$ be a standard Brownian motion. For each $$t\in [0,1]$$, it is known that, e.g. from Burkholder-Davis-Gundy's inequality

$$\mathbb E\big[\sup_{s\in [t,t+\Delta t]}|W_s-W_t|^p\big]=O(\Delta t^{p/2}),\quad \forall p\ge 1,$$

where $$O$$ refers to "of order". Do we have an estimate of

$$\mathbb E\big[\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\big]?$$

A student asked me this question when I taught Euler's scheme applied to SDEs and its convergence, but I cannot find any reference. Any answers, comments or references are appreciated.

• Lévy's modulus of continuity theorem might be a good start. Aug 6, 2021 at 20:25
• Judging be the abstract, On the Moments of the Modulus of Continuity of Itô Processes by M. Fischer and G. Nappo should either contain an answer or a reference to the right paper. Aug 6, 2021 at 20:28
• Thank you very much for your kindness. This is not the first time that I have your help. Thanks a lot! Aug 6, 2021 at 20:42
• @MateuszKwaśnicki Thank you very much for your kindness. This is not the first time that I have your help. Thanks a lot! Btw, do you have ideas on the question posted at mathoverflow.net/questions/398321/… ? Aug 6, 2021 at 20:53

For $$n\in\mathbb{Z}_{\geq 0}$$ and $$0\leq i< 2^n$$, denote
$$X_{n,i}=\sup_{t\in[i2^{-n}, (i+1)2^{-n}]}{|W_t-W_{i2^n}|}.$$
Let $$n$$ be such that $$2^{-{n}}<|\Delta t|\leq 2^{-{n+1}}$$. Then, $$\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\leq 4\sup_{i}|X_{i,n}|^p,$$ because such $$s$$ and $$t$$ must belong to a union of three consecutive diadic intervals of length $$2^{-n}$$. For a fixed $$n$$, $$X_{n,i}$$ are i. i. d. random variables; moreover, by Brownian scaling, $$X_{n,i}\stackrel{\mathcal{D}}{=}2^{-n/2}X_{0,0}.$$ If $$F(t)$$ denotes the cumulative disctribution function of $$X_{0,0}$$, then we have, with $$N=2^n,$$ $$\mathbb{P}(\sup_{i}|X_{i,n}|\leq t)=F(N^{1/2} t)^{N}.$$ It is well known that $$\mathbb{P}(\sup_{[0,1]} W_t>a)=2\mathbb{P}(W_t>a)\leq 2e^{-a^2/2}$$ for $$a$$ large enough, so that $$F(t)\geq 1-4e^{-t^2/2}$$, and $$\mathbb{P}(\sup_{i}|X_{i,n}|>t)\leq 1-(1-4e^{-Nt^2/2})^{N}.$$ Let $$T_n:=\sqrt{2\log 2}n^\frac12 N^{-\frac12}$$, then, plugging into the above, $$\mathbb{P}(\sup_{i}|X_{i,n}|>kT_n)\leq 1-e^{N\log(1-4\cdot 2^{-k^2n})}\leq 10\cdot 2^{(1-k^2)n}.$$ That is to say, the median of $$\sup_{i}|X_{i,n}|$$ is of order at most $$T_n$$ and its tail decays very sharply after that, implying
$$\mathbb{E}\sup_{i}|X_{i,n}|^p=O(T_n^p)$$ for all $$p$$, or $$\mathbb{E}\left(\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\right)=O\left((|\Delta t\log (\Delta t)|^\frac{p}{2}\right).$$