# Questions tagged [gaussian-measure]

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.

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### Gaussian concentration to upper bound the variance of the norm of a gaussian vector

I want to prove the following: Let $X_1$ be a centered Gaussian vector in $\mathbb{R}^d$ with covariance matrix $\Sigma$. Then: $Var( \|X_1 \|) \lesssim \left(\mathbb{E} \|X_1 \| \right)^2$ This is ...
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### 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 ...
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### Find larger space such that identity covariance is trace class

Assume I have the identity covariance operator $C = \mathrm{Id}$ on the Sobolev space $H^1([0,1]^2)$. Can I find a larger space $H$ with $H^1([0,1]^2) \subset H$ such that this covariance $C$ is trace ...
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### Definition of stochastic differential equations in infinite dimensions without traceclass

In infinite dimensions, the main idea to define a stochastic differential equation is to consider a trace class (cylindrical) Brownian motion. However, I've been wondering whether there exists a body ...
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### 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 ...
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### Geometry of Voronoi cells of $n+1$ equidistant points in $\mathbb{R}^n$

These questions come from the reading of the following article: Milman, E., & Neeman, J. (2022). The Gaussian Double-Bubble and Multi-Bubble Conjectures. Annals of Mathematics, 195(1), 89-206. (...
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I define a Gaussian measure on a separable Banach space $(\mathcal{F},\|\cdot\|)$ as in van der Vaart and van Zanten (2008): A Borel measurable random element $W$ with values in a separable Banach ...