Questions tagged [wiener-measure]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
72 views

What is the justification for using Wiener integrals to integrate over a space of differentiable functions?

In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
user avatar
6 votes
2 answers
788 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 ...
user avatar
0 votes
2 answers
694 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 ...
user avatar
  • 697
2 votes
0 answers
87 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,...
user avatar
  • 475
0 votes
2 answers
287 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 \...
user avatar
3 votes
1 answer
181 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 ...
user avatar
0 votes
1 answer
112 views

"Geometric" Decomposition of Wiener Space

Let $C_0([0,1];\mathbb{R}^d)$ be the classical Wiener space (of continuous paths with initial value $0$) and let $\nu$ be the Wiener measure on this space. Does there exist a countable family $\left\{...
user avatar
  • 4,997
1 vote
1 answer
208 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 ...
user avatar
  • 4,997
2 votes
1 answer
100 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 ...
user avatar
  • 235
0 votes
0 answers
127 views

On measurability in Wiener space

Let $f$ be a complex-valued continuous function on Wiener space such that $|f|$ is measurable. Is $f$ then measurable, too? I am looking for a proof or a counterexample.
user avatar
0 votes
1 answer
117 views

Convergence of an integral with respect to the Wiener measure

Most probably this question should be well studied in the theory of stochastic processes, but I am not educated in that area. Sorry if this question is too elementary. Let $V\colon \mathbb{R}\to \...
user avatar
  • 19k
5 votes
2 answers
415 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 ...
user avatar
1 vote
1 answer
600 views

interpretation of the transition probability of a brownian motion in terms of the Wiener measure

Let $W(t)$ be a standard brownian motion in $E \triangleq \mathbb{R}^d$. The transition probability from a state $x \in E$ at time $t$ to a state $y \in E$ at time $T$ is $$ p(x,t;y,T) = \frac{1}{\...
user avatar
  • 181
1 vote
2 answers
219 views

Conditional Wiener measure continuous

consider a complete Riemannian manifold $M$ with heat kernel $p_M$ and let $U\subset M$ be an open set. Let $W_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ...
user avatar
3 votes
1 answer
866 views

can I integrate product or square of a white noise in any sense?

Assume that we have $\epsilon_1, \; \epsilon_2$ independent white noises. Can I write $\int_{0}^1 \epsilon_1^2(t)dt$ Can I write $\int_{0}^1 \epsilon_1(t) \epsilon_2(t)dt$ 1 and 2 obviously make no ...
user avatar
1 vote
0 answers
149 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 ...
user avatar
  • 1,254