# The Wiener measure of an open set

There is so much written about the Brownian motion and I suspect the answers to the questions below are hidden in somewhere in the literature but I cannot find them

Denote by $$E$$ the Banach space of continuous functions $$\newcommand{\bR}{\mathbb{R}}\newcommand{\ve}{\varepsilon}\newcommand{\bsW}{\boldsymbol{W}}$$ $$f:[0,1]\to\bR$$ such that $$f(0)=0$$. The norm on $$E$$ is the sup-norm.

The Wiener measure defines a Borel measure $$\bsW$$ on $$E$$. Let $$f_0\in E_0$$ and $$\ve>0$$. We set $$w(f_0,\ve):=\bsW\Big[\;\big\{\; f\in E;\;\;\Vert f-f_0\Vert<\ve\,\big\}\;\Big].$$

Question 1. Is it true that $$w(f_0,\ve)>0$$ for all $$f_0\in E$$ and $$\ve>0$$?

Question 2. Can one produce an explicit positive lower bound for $$w(f_0,\ve)$$ in terms of $$\ve>0$$ and the modulus of continuity of $$f$$? In particular, if $$f_0$$ is Lipschitz, can one produce a lower bound in terms of $$\ve>0$$ and the Lipschitz constant of $$f_0$$?

For Question 1 I have an argument based on Cameron-Martin formula? Is there any other more "elementary" argument?

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• According to this question on Math.SE, this is called "small ball estimates". Have you tried searching under this term? Commented Jan 16, 2022 at 19:37
• I've seen something like this in Peres-Morters book Exercise 1.8, where they first built a dyadic function that is epsilon close to f0 and then continue the Levy construction of adding Gaussians in-between to built a Brownian motion. Commented Jan 16, 2022 at 19:39

$$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\om}{\omega}$$Let $$g:=f_0$$. There is some real $$\de>0$$ such that $$\begin{equation*} \om(g,\de):=\max\{|g(y)-g(x)|\colon x,y\in[0,1], |y-x|\le\de\}<\ep/3<\ep/2. \end{equation*}$$ Take now any $$t_0,\dots,t_n$$ such that $$0=t_0<\dots and $$\begin{equation*} \de_k:=t_k-t_{k-1}<\de \end{equation*}$$ for all $$k\in[n]:=\{1,\dots,n\}$$.

If $$f\colon[0,1]\to\mathbb R$$ is such that for all $$k\in[n]$$ and all $$t\in[t_{k-1},t_k]$$ we have $$f(t)\in I_k:=[g(t_k)-\ep/2,g(t_k)+\ep/2]$$, then $$\|f-g\|<\ep$$.

So, for any standard Wiener process $$W$$, $$\begin{equation*} w(g,\ep)=P(\|W-g\|<\ep)\ge p:=p_1\cdots p_n, \end{equation*}$$ where $$\begin{equation*} p_k:=\min_{x\in J_{k-1}}P(x+W_t\in I_k\ \forall t\in[0,\de_k], x+W_{\de_k}\in J_k), \end{equation*}$$ $$J_0:=[0,0]=\{0\}$$, $$J_n:=I_n$$, and $$J_k$$ is the closed interval that is the middle third of the interval $$I_k\cap I_{k+1}$$ for $$k\in[n-1]$$. Note that for $$k\in[n-1]$$ the length of the interval $$I_k\cap I_{k+1}$$ is $$>\ep/2$$ and hence the length of the interval $$J_k$$ is $$>\ep/6>0$$. Also, for each $$k\in[n]$$ the shortest distance from any point $$x\in J_{k-1}$$ to the set $$\{g(t_k)-\ep/2,g(t_k)+\ep/2\}$$ of the endpoints of the interval $$I_k$$ is $$>\ep/6>0$$. So, $$p_k>0$$ for all $$k\in[n]$$, and hence $$p>0$$. Moreover, one can give an explicit expression of each $$p_k$$ in terms of $$\de_k$$, $$I_k$$, $$J_k$$, $$J_{k-1}$$ -- cf. e.g. Proposition 6.10.6, p. 533.

Thus, one can explicitly express the lower bound $$p>0$$ on $$w(g,\ep)$$ in terms of $$\ep$$, $$t_1,\dots,t_n$$, and $$g(t_1),\dots,g(t_n)$$.

For $$g(t)=t (5 - 18 t + 12 t^2)$$, $$\ep=5/2$$, $$n=4$$, $$\de_k=1/4$$ for $$k\in[4]$$, the picture below shows the graphs $$\{(t,g(t))\colon t\in[0,1]\}$$ (blue), $$\{(t,g_-(t))\colon t\in[0,1]\}$$ (gold), $$\{(t,g_+(t))\colon t\in[0,1]\}$$ (green), and a path of the Wiener process (gray) belonging to the event \begin{equation*} \begin{aligned} B&:=\{g_-(t) where $$g_\pm(t):=g(t_k)\pm\ep/2$$ for $$t\in(t_{k-1},t_k]$$, $$k\in[4]$$. We have $$P(\|W-g\|<\ep)\ge P(B)\ge p>0$$.

• Thank you very much. Commented Jan 17, 2022 at 10:13

This is known as the support theorem for Brownian motion. Besides the proof in the answer of Iosif Pinelis and the proof in Exercise 1.8 of [1], there is also a proof on page 59 of [2]. Generalizations are discussed in [3]-[5].

[1] Mörters, Peter, and Yuval Peres. Brownian motion. Vol. 30. Cambridge University Press, 2010. https://yuvalperes.com/brownian-motion/

[2] R. Bass, Probabilistic Techniques in Analysis, Springer, New York (1995). MR1329542

[5] Stroock, Daniel W.; Varadhan, S. R. S. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 333–359. Univ. California Press, Berkeley, Calif., 1972.

• Thank you very much. Commented Jan 17, 2022 at 10:13