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59 views

Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?

The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
1 vote
0 answers
70 views

On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$

Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
2 votes
1 answer
199 views

Gaussian Poincare inequality in $1$ dimensions together with localization issue

Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$. Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$. Then, for any smooth mapping $f : \...
2 votes
0 answers
62 views

On a real smooth version of white noise distribution theory

In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
5 votes
1 answer
284 views

Malliavin derivative of stopped Brownian motion

Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion" I have a small question concerning the Malliavin derivatives. It could ...
5 votes
1 answer
2k views

Definition of infinite-dimensional Gaussian random variable

For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this: Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random variable $u \in H$ is ...
1 vote
0 answers
177 views

A question on Gaussian small ball probability

Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$ where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...
2 votes
1 answer
274 views

Small ball Gaussian probabilities with moving center

I would like to prove (if possible, otherwise find a counterexample for) the following lemma: Let $(X,\|\cdot \|_X)$ be a separable Banach space. Additionally, we have a centred Gaussian measure $\mu$ ...
3 votes
0 answers
145 views

Density of invariant measure of stochastic differential equation

I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
1 vote
0 answers
206 views

The quadratic variation of $\int_0^t\int_T^Sg(s,x) \, dW_s^x \, dx$

Consider the process $W^x_t$ which is a Brownian motion for every $x\geq 0$ such that $$d\langle W_t^x,W_t^y\rangle=Q(x,y)\,dt$$ where $Q$ is some non-negative definite function. Now consider the ...
2 votes
1 answer
173 views

Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?

It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$, $$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
1 vote
1 answer
82 views

Local inverse bound of Cameron Martin and Banach norms

Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
4 votes
0 answers
322 views

Compactness of semigroups of one-dimensional diffusions

I have a question about semigroups of one-dimensional diffusions. Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as $$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$ It is ...
1 vote
0 answers
56 views

About a class of expectations

Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
3 votes
0 answers
569 views

Domain of the Generator of a Bessel process

Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$ \begin{align} \rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t} \end{align} where $(W_{t})_{t\geq ...
1 vote
0 answers
100 views

Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion

Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$: $$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
7 votes
1 answer
624 views

Expectation involving maximum of Gaussian variables

Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
4 votes
1 answer
225 views

Multivariate Zero-Bias Transform

The zero-bias transform for a univariate random variable $W$ is defined as a random variable $W^*$ satisfying \begin{align} \mathbb{E} [ W \cdot f(W )] = \mathbb{E} [ f' (W^*)] \end{align} for any ...
2 votes
1 answer
338 views

complex version of Gaussian integral

in DaPrato/Zabczyk's book "Second Order Partial Differential Equations in Hilbert Spaces", there is a useful proposition (Prop. 1.2.8) about a particular calculation of a Gaussian integral in Hilbert ...
3 votes
0 answers
78 views

Perscribed/Inverting Conditional Expectation

I'm having difficulty finding papers which deal with the following inversion problem. Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
4 votes
0 answers
414 views

Definition of the Stratonovich integral in Hilbert spaces

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a (standard, real-...
6 votes
2 answers
748 views

Does there exist a stochastic time derivative?

The Setup Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE $$ dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t $$ and $f$ is a smooth function. My Question Is there a ...
5 votes
2 answers
673 views

Regular Dirichlet form and the associated transition kernel

I am reading a paper by Fukushima "On a stochastic calculus related to Dirichlet forms and distorted Brownian motions" and support it by a book "Dirichlet forms and symmetric Markov processes" by ...
3 votes
1 answer
164 views

Representation of support of Gaussian measure by kernels of no-variance functionals

Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for $$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$ ...