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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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    $\begingroup$ According to this question on Math.SE, this is called "small ball estimates". Have you tried searching under this term? $\endgroup$ Jan 16 at 19:37
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    $\begingroup$ 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. $\endgroup$ Jan 16 at 19:39

2 Answers 2

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$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\om}{\omega}$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 $0=t_0<\dots<t_n=1$ 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)<W_t<g_+(t)\ \forall t\in[0,1], W_{t_k}\in J_k\ \forall k\in\{0\}\cup[n]\} \\ &\subseteq\{\|W-g\|<\ep\}, \end{aligned} \end{equation*} where $g_\pm(t):=g(t_k)\pm\ep/2$ for $t\in(t_{k-1},t_k]$, $k\in[4]$. We have $P(\|W-g\|<\ep)\ge P(B)\ge p>0$.

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  • $\begingroup$ Thank you very much. $\endgroup$ Jan 17 at 10:13
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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

[3] https://fabricebaudoin.wordpress.com/2013/04/10/lecture-30-the-stroock-varadhan-support-theorem/

[4] http://www.numdam.org/article/SPS_1994__28__36_0.pdf

[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.

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  • $\begingroup$ Thank you very much. $\endgroup$ Jan 17 at 10:13

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