# Questions tagged [wiener-measure]

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### How does the Wiener measure work at the intersection of cylinders?

For any integer $n$, any choice of $0 < t_1 < \cdots < t_n \leq 1$, and any Lebesgue measurable set, $E \in \mathbb{R}^n$ define the “cylinder”
$$I = I(n;t_1;\cdots;t_n;E) := \{ \beta(·) \in ...

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### 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. ...

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### How does the conditional Wiener measure work?

In the theorem below $P_D$ means the heat kernel in the open $D \subset \mathbb{R}^m$ and $P_m$ is the heat kernel in whole $\mathbb{R}^m.$
I know absolutely nothing about what Brownian bridges are, ...

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### 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(...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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,...

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### 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 \...

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### 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 ...

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### "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\{...

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### 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 ...

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### 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 ...

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### 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.

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### 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 \...

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### 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 ...

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### 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}{\...

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### 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 ...

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### 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 ...

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### 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 ...