About measures on infinite-dimensional topological vector spaces for which every continuous linear functional is Gaussian. Also in conjunction with abstract Wiener spaces and Gaussian processes.
Questions tagged [gaussian-measure]
42 questions
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How to apply Kolmogorov's continuity criterion to a Karhunen–Loève expansion
Disclaimer: this post can now also be found on mathoverflow, see here.
I would like to understand the proof of the following theorem, which is a simpler version of Corollary 4.24 from Martin Hairer's lecture notes on stochastic PDEs, see here. The…
hannes
- 91
5
votes
1 answer
Abstract Wiener Space for $\ell^2(\mathbb{R})$
Let $\ell^2(\mathbb{R})$ denote the space of square-summable real-valued sequences equipped with the inner product
$$ \langle x, y \rangle = \sum_{n = 1}^{\infty} x_n y_n $$
Let $\nu$ denote its canonical cylinder measure i.e. a cylinder measure…
G. Chiusole
- 5,594
4
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Why are Gaussian measures seen as the standard measure for infinite dimensional spaces?
I'm learning about infinite dimensional probability, and most resources I've consulted so far motivate things by saying there is no infinite dimensional Lebesgue/translation invariant measure that gives nontrivial measures for unit balls in a Banach…
CBBAM
- 7,149
4
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1 answer
Restriction of Gaussian measure
In these lecture notes, exercise 3.39 is posed:
Let $\tilde B, B$ be Banach spaces, and $\mu$ be a Gaussian measure on $B$. If $\tilde B$ is continuously embedded into $B$ and $\mu(\tilde B)=1$, then the "restricted" measure $\tilde \mu$ on $\tilde…
SoupMath
- 23
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Energy Distance between Multivariate Gaussian Distributions
The square of energy Distance between CDFs $F$ and $G$, of $X$ and $Y$ resp., is defined here as
$$d^2(F, G) = E||X-Y|| - E||X-X'|| - E||Y-Y'||$$
where $(X, X')$ and $(Y, Y')$ are IID pairs.
I am looking for an explicit form for multivariate…
cybershiptrooper
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4
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How does the Cameron-Martin space of a Gaussian Markov process with initial condition $0$ change when we consider a random initial condition
Suppose we are given a real-valued Gaussian Markov process with initial condition zero
$$Y_t = \int_0^t f(s) dW_s \quad t \in [0,T], f\in L^2([0,T].$$
Following [Example 4.5, 1], $Y$ is distributed according to Gaussian measure on the canonical path…
Lance
- 1,136
4
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1 answer
Reference: Covariance operator is compact
I am looking for a reference for the following theorem:
Let $(X,\Vert \cdot \Vert)$ be a separable Banach space and $C: X^{\ast} \rightarrow X \subseteq X^{\ast \ast}, f \mapsto \int_X f(x) \cdot - ~ d\mu(x) =: q(f, -)$ be the covariance operator…
G. Chiusole
- 5,594
3
votes
1 answer
Show $Y_t = X(f\mathbf 1_{[0,t]})$ admits a continuous modification if $f\in L^{2+\epsilon}_{\mathrm{loc}}$
I am stuck on the following exercise (this is one of the first exercises in Revuz and Yor, so the solution should be from first principles).
Let $f\in L^{2+\epsilon}_{\mathrm{loc}}(\mathbb R_+,dx)$ and put $Y_t = X(f\mathbf 1_{[0,t]})$, where $X$ is…
Andrew
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A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$
Let $\mathcal{E}(S^1)$ be the space of smooth functions on the circle $S^1$ and denote its dual as $\mathcal{E}'(S^1)$.
Then, by the Minlos theorem, there exists a unique probability measure $\mu$ on $\mathcal{E}'(S^1)$ such…
Keith
- 8,359
3
votes
0 answers
Holder-continuity of the fractional Brownian motion in a compact $[0,T]$ and dependence in $T$
I know that the fractional Brownian motion has a Holder-continuous version (thanks to Kolmogorov's continuity theorem for example). Basically, for any $T>0$, $\epsilon>0$ and $t,s \in [0,T]$, there exists a random variable, say $C_T$ such that:
$$…
El Mehdi
- 39
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Support of the Fractional Stationary Ornstein Uhlenbeck process (first kind)
I am interested in the support of the fractional stationary Ornstein Uhlenbeck process (first kind). In particular, I want to know whether smooth functions (or even only smooth functions starting at 0) are in the support. To put it into some…
Chris
- 66
2
votes
2 answers
Path integral formalization using measure-based integration and equivalence with limit-based definitions
This question is coming from a computational physicist who is very comfortable with numerical and computational math, but much less comfortable with distributions, measures, and Lebesgue integration. The context of this question is that I'm trying…
2
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1 answer
Help me understand this proof of "the covariance of a Gaussian measure is trace-class"
So I am reading an introductory script on stochastic analysis in Hilbert spaces and there is a step in the proof of "Gaussian measures have trace-class covariance" that I don't understand:
We are working with a Gaussian measure $\mu$ on a separable…
Dasi
- 279
2
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1 answer
Proof for gaussian random variables
I am trying to prove the KK inequality. I tried to do it on my own but could not manage to work out the full details and unfortunately, it seems like it is not present on the web any proof of the theorem.
In the following, there's the…
Angelo
- 35
2
votes
1 answer
Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and invertible operator.
Let $u\in H$ and $n\in \mathbb{N}$,…
John
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