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Results tagged with wiener-measure
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user 36721
4
votes
Accepted
What exactly is the relation between the Wiener process and Wiener measure?
The Wiener measure $w$ is the distribution of the Wiener process/random function $W$ on $C[0,1]$; that is,
$$P(W\in A)=w(A)$$
for all Borel sets $A\subseteq C[0,1]$. Here "Borel sets" can be replaced …
- 128k
1
vote
Accepted
"Geometric" Decomposition of Wiener Space
For natural $n$, let
$$U_n:=\{x=(x_1,\dots,x_d)\in C_0([0,1];\mathbb{R}^d)\colon \Phi(x_1(1))\in\delta_n\},
$$
where $\Phi$ is the standard normal pdf and $\delta_n:=(1-1/2^{n-1},1-1/2^n)$. Then the …
- 128k
10
votes
Accepted
The Wiener measure of an open set
$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\om}{\omega}$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], …
- 128k
4
votes
Accepted
How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(...
The convergence of the discretized version $\max_{x \in \{0, \ldots, N\}} |W(x/N)|$ of $M:=\max_{0 \le t \le 1} |W(t)|$ to $M$ will be very slow -- at the rate of $1/\sqrt N$, according to Korolyuk 19 …
- 128k