11
votes
The Wiener measure of an open set
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]. ...
- 14.2k
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,...
- 128k
4
votes
Wiener Measure measure on functions?
The trick is to regard the Wiener measure as a random sample function $f(x,t)$ where $x\in (\Omega, \mathscr{F},P)$ and $t\in \mathscr{T}$ is the time index set. Then the whole stochastic process can ...
- 8,071
4
votes
Accepted
How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?
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 ...
- 128k
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 ...
- 128k
3
votes
Accepted
Is there a generalised version of the Donsker invariance principle for a "sort-of continuous-time-random-walk"?
I think Iosif Pinelis is correct, but his comment should be expanded as follows.
Notation: let
$$ p(t) = \lfloor t + 1\rfloor - t , \qquad q(t) = t - \lfloor t\rfloor . $$
Whenever we have a discrete-...
- 17.2k
3
votes
Accepted
How does the conditional Wiener measure work?
Brownian bridges, heuristically, are Brownian motions that are conditioned to start at a given point and end at a given point.
In this case, the conditional Wiener measure they refer to is just the ...
- 6,155
3
votes
Accepted
Onsager-Machlup functional when drift is time-dependent
Using the argument of http://users.sussex.ac.uk/~md326/MAP.pdf or https://arxiv.org/abs/2209.04523
We have that if $\mu_0$ is a centered Gaussian measure then its Onsager-Machlup function is $\...
- 1,904
3
votes
Accepted
Reference Request: 2-Wasserstein Metric on Wiener Space
This is discussed thoroughly in the following reference; see Section 3 and specifically Theorem 3.5.
Gelbrich, Matthias, On a formula for the $L^2$ Wasserstein metric between measures on Euclidean ...
- 6,045
3
votes
Accepted
can I integrate product or square of a white noise in any sense?
A consistent framework for "nonlinear stochastic calculus" has been developed in The square of white noise as a Jacobi field (2004), building on earlier work in Squared white noise and other non-...
- 188k
2
votes
Wiener Measure measure on functions?
It seems that what you are looking for is the notion of "abstract Wiener space", which provides a rigorous setting for putting a white noise on various spaces of functions. Long story short, you ...
- 2,135
2
votes
What exactly is the relation between the Wiener process and Wiener measure?
We have to differentiate between the probability measure $w$ as a measure on the space $(C[0,1],\mathcal{B}(C[0,1]))$, the Wiener measure and the concept of an Ito integral $\int_{[0,t]} X_s dW_s$, $0 ...
- 2,101
1
vote
Functional integral formulas for the wave equation and other hyperbolic PDEs
See also a preprint https://arxiv.org/abs/1306.2382 and a paper https://repository.lsu.edu/josa/vol5/iss2/3/ which use a kind of "Wick rotation." In the Chatterjee preprint, a Cauchy random ...
- 1,904
1
vote
Abstract Wiener spaces for pinned processes (e.g., Brownian Bridge)
A Gaussian process can be thought of as a "standard Gaussian" on its Cameron-Martin space $\mathcal{M}$. That is, given an orthonormal basis $\psi_1,\psi_2, \dots$ of the Cameron-Martin ...
- 8,992
1
vote
Accepted
Brownian level sets and continuous functions
Call two sequences $(a_n)$ and $(b_n)$ tail-equivalent if there are $p$ and $q$ such that $a_{p+n} = b_{q+n}$ for every $n \geqslant 0$. Write $W(t)$ rather than $W_t$.
Suppose that $f$ with the ...
- 17.2k
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
1
vote
Accepted
Convergence of an integral with respect to the Wiener measure
The conditional Wiener measure is concentrated on the space $C(L,a,b)$ of continuous curves $x : [0,L] \to \mathbb R$ such that $x(0) = a$ and $x(L) = b$, endowed with the topology given by the ...
- 5,407
1
vote
interpretation of the transition probability of a brownian motion in terms of the Wiener measure
I have find in [equation 1.1.51, page 24,Chaichian M, Demichev A. Path integrals in physics, vol I : stochastic processes and quantum mechanics. Bristol (Philadelphia): Institute of Physics Publishing;...
- 365
1
vote
Accepted
Conditional Wiener measure continuous
As the transition densities for the Wiener measure are continuous, weak continuity results of a type that may be what you seek can be found in http://projecteuclid.org/euclid.aop/1298669175 ["...
- 1,989
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