Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains

Abstract

Let $X$ be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure $\gamma$ and let ${\lambda }_1$ be the maximum eigenvalue of the covariance operator associated with $\gamma$. The associated Cameron--Martin space is denoted by $H$. For a sufficiently regular convex function $U:X\to \mathbb{R}$ and a convex set $\Omega \subseteq X$, we set $\nu :=e^{-U}\gamma$ and we consider the semigroup $(T_{\Omega }(t){)}_{t\ge 0}$ generated by the self-adjoint operator defined via the quadratic form \[ (φ,ψ)\mapsto \int_Ω\langle D_Hφ,D_Hψ\rangle_Hdν, \] where $\phi ,\psi$ belong to $D^{1,2}(\Omega ,\nu )$, the Sobolev space defined as the domain of the closure in $L^2(\Omega ,\nu )$ of $D_H$, the gradient operator along the directions of $H$. A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_{\Omega }(t){)}_{t\ge 0}$. In particular, we show that \[ |D_H T_Ω(t)f|_H^p\le e^{- p λ_1^{-1} t}(T_Ω(t)|D_H f|^p_H), \qquad\, t>0,\ ν\textrm{ -a.e. in }Ω, \] for any $p\in [1,+\infty )$ and $f\in D^{1,p}(\Omega ,\nu )$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in $\Omega$ for the measure $\nu$ and some improving summability properties for $(T_{\Omega }(t){)}_{t\ge 0}$. In addition we prove that if $f$ belongs to $L^p(\Omega ,\nu )$ for some $p\in (1,\infty )$, then \[|D_H T_Ω(t)f|^p_H \leq K_p t^{-\frac{p}{2}} T_Ω(t)|f|^p,\qquad \, t>0,\ ν\text{-a.e. in }Ω,\] where $K_p$ is a positive constant depending only on $p$. Finally we investigate on the asymptotic behaviour of the semigroup $(T_{\Omega }(t){)}_{t\ge 0}$ as $t$ goes to infinity.

References in corpus (3)

• Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space
• Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure
• Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: the convex subset case

Cited by in corpus (6)

• Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces
• Sobolev spaces with respect to a weighted Gaussian measures in infinite dimensions
• On functions of bounded variation on convex domains in Hilbert spaces
• Regarding the domain of non-symmetric and, possibly, degenerate Ornstein--Uhlenbeck operators in separable Banach spaces
• $L^2$-theory for transitions semigroups associated to dissipative systems
• Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations