All Questions
Tagged with wiener-measure pr.probability
12 questions
3
votes
2
answers
401
views
Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
- 11.8k
0
votes
1
answer
211
views
Abstract Wiener spaces for pinned processes (e.g., Brownian Bridge)
In introductions to abstract Wiener spaces, the sample paths usually form a Banach space; so, in particular, the sum of two sample paths is a valid sample path and also an element of the Banach space. ...
- 101
7
votes
1
answer
249
views
Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...
- 203
8
votes
2
answers
1k
views
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 ...
- 34.7k
5
votes
2
answers
3k
views
What exactly is the relation between the Wiener process and Wiener measure?
The Wiener measure is (in the classical sense) a Gaussian measure on the Banach space $C[0,1]:=\{f:[0,1] \to \mathbb{R} \mid f\text{ is continuous and } f(0)=1\}$.
The Wiener process is a stochastic ...
- 3,477
2
votes
0
answers
138
views
On the difference between Malliavin derivative and Gross-Sobolev derivative
Let $W=C_0([0,1],\mathbb R^d)$ be the classical Wiener space equipped with $\mu$ the Wiener measure.
If $F:W\to\mathbb R$ is a cylindrical function of the form
\begin{align*}
F(w)=f(W_{t_1}(w),\cdots,...
- 515
1
vote
2
answers
594
views
Is there a generalised version of the Donsker invariance principle for a "sort-of continuous-time-random-walk"?
(The following question arises from my Math.SE question https://math.stackexchange.com/questions/3643865.)
Let $\rho$ be a probability measure on $\mathbb{R} \times (0,\infty)$, and writing $\ \pi_1 \...
- 698
3
votes
1
answer
232
views
Brownian level sets and continuous functions
Let $V_t$ and $W_t$ be independent standard Wiener processes ($t\ge 0$, $W_t,V_t\in\mathbb R$).
Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$,
$$
W_t=W_s\iff ...
- 24.8k
2
votes
1
answer
257
views
Reference Request: 2-Wasserstein Metric on Wiener Space
Suppose that X is the subspace of the set of probability measures on the classical Wiener space $C[0,T]$, for some $T>0$, comprised of Gaussian measures.
In the finite-dimensional setting, the ...
- 5,407
2
votes
1
answer
154
views
How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?
How far away is
$$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$
from
$$\max_{0 \leq t \leq 1} |W(t)|$$
In other words, if you simulate a Wiener process over a finite ...
- 277
6
votes
2
answers
698
views
Wiener Measure measure on functions?
I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated ...
1
vote
0
answers
192
views
References about distances between singular probability measures
I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total ...
- 1,334