We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation $v$ of the Ornstein--Uhlenbeck operator $L$ possesses a highly integrable density with respect to the Gaussian measure satisfying the n
on-perturbed equation provided that $v$ is sufficiently integrable. More generally, a similar estimate is proved for solutions to inequalities connected with Markov semigroup generators under the curvature condition $CD(theta,infty)$. For perturbations from $L^p$ an analog of the Log-Sobolev inequality is obtained. It is also proved in the Gaussian case that the gradient of the density is integrable to all powers. We obtain dimension-free bounds on the density and its gradient, which also covers the infinite-dimensional case.
In this paper we study the regularity properties of linear and polynomial images of Skorohod differentiable measures. Firstly, we obtain estimates for the Skorohod derivative norm of a projection of a product of Scorohod differentiable measures. In t
he second part of the paper we prove Nikolskii--Besov regularity of a polynomial image of a Skorohod differentiable measure on $mathbb{R}^n$.
Let $gamma$ be the standard Gaussian measure on $mathbb{R}^n$ and let $mathcal{P}_{gamma}$ be the space of probability measures that are absolutely continuous with respect to $gamma$. We study lower bounds for the functional $mathcal{F}_{gamma}(mu) =
{rm Ent}(mu) - frac{1}{2} W^2_2(mu, u)$, where $mu in mathcal{P}_{gamma}, u in mathcal{P}_{gamma}$, ${rm Ent}(mu) = int logbigl( frac{mu}{gamma}bigr) d mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. The minimizers of $mathcal{F}_{gamma}$ are solutions to a dimension-free Gaussian analog of the (real) Kahler-Einstein equation. We show that $mathcal{F}_{gamma}(mu) $ is bounded from below under the assumption that the Gaussian Fisher information of $ u$ is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.
We give a new description of classical Besov spaces in terms of a new modulus of continuity. Then a similar approach is used to introduce Besov classes on an infinite-dimensional space endowed with a Gaussian measure.
We give a new characterization of Nikolskii-Besov classes of functions of fractional smoothness by means of a nonlinear integration by parts formula in the form of a nonlinear inequality. A similar characterization is obtained for Nikolskii-Besov cla
sses with respect to Gaussian measures on finite- and infinite-dimensional spaces.
We prove that the distribution density of any non-constant polynomial $f(xi_1,xi_2,ldots)$ of degree $d$ in independent standard Gaussian random variables $xi$ (possibly, in infinitely many variables) always belongs to the Nikolskii--Besov space $B^{
1/d}(mathbb{R}^1)$ of fractional order $1/d$ (and this order is best possible), and an analogous result holds for polynomial mappings with values in $mathbb{R}^k$. Our second main result is an upper bound on the total variation distance between two probability measures on $mathbb{R}^k$ via the Kantorovich distance between them and a suitable Nikolskii--Besov norm of their difference. As an application we consider the total variation distance between the distributions of two random $k$-dimensional vectors composed of polynomials of degree $d$ in Gaussian random variables and show that this distance is estimated by a fractional power of the Kantorovich distance with an exponent depending only on $d$ and $k$, but not on the number of variables of the considered polynomials.
We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting cons
equences. Our second result gives a characterization of limits in law for sequences of such vectors.
