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 $$U:X\rightarrow {{\mathbb {R}}}$$ and a convex set $$\Omega \subseteq X$$ , we set $$\nu :=\hbox {e}^{-U}\gamma $$ and we consider the semigroup $$(T_\Omega (t))_{t\ge 0}$$ generated by the self-adjoint operator defined via the quadratic form $$\begin{aligned} (\varphi ,\psi )\mapsto \int _\Omega {\left\langle D_H\varphi ,D_H\psi \right\rangle }_H \hbox {d}\nu , \end{aligned}$$ where $$\varphi ,\psi $$ 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 $$\begin{aligned} |D_H T_{\Omega }(t)f|_H^p\le \hbox {e}^{- p \lambda _1^{-1} t}(T_{\Omega }(t)|D_H f|^p_H), \quad \, t>0, \nu \text {-a.e. in }{\Omega }, \end{aligned}$$ 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 Poincare 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 $$\begin{aligned} |D_H T_\Omega (t)f|^p_H \le K_p t^{-\frac{p}{2}} T_\Omega (t)|f|^p,\quad \, t>0, \nu \text {-a.e. in }\Omega , \end{aligned}$$ 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.