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

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 $\mathrm{\Omega }\subseteq X$, we set $\nu :=e^{-U}\gamma$ and we consider the semigroup $(T_{\mathrm{\Omega }}(t){)}_{t\ge 0}$ generated by the self-adjoint operator defined via the quadratic form $$(\phi ,\psi )\mapsto {\int }_{\mathrm{\Omega }}⟨D_H\phi ,D_H\psi {⟩}_{H}d\nu ,$$ where $\phi ,\psi$ belong to $D^{1,2}(\mathrm{\Omega },\nu )$, the Sobolev space defined as the domain of the closure in $L^2(\mathrm{\Omega },\nu )$ of $D_H$, the gradient operator along the directions of $H$. A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_{\mathrm{\Omega }}(t){)}_{t\ge 0}$. In particular, we show that $$\mathrm{\mid }{D}_H{T}_{\mathrm{\Omega }}(t)f{\mathrm{\mid }}_H^p\le e^{-p{\lambda }_1^{-1}t}(T_{\mathrm{\Omega }}(t)\mathrm{\mid }{D}_{H}f{\mathrm{\mid }}_H^p),t>0,\nu \text{ -a.e. in }\mathrm{\Omega },$$ for any $p\in [1,+\mathrm{\infty })$ and $f\in D^{1,p}(\mathrm{\Omega },\nu )$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincar'e inequality in $\mathrm{\Omega }$ for the measure $\nu$ and some improving summability properties for $(T_{\mathrm{\Omega }}(t){)}_{t\ge 0}$. In addition we prove that if $f$ belongs to $L^p(\mathrm{\Omega },\nu )$ for some $p\in (1,\mathrm{\infty })$, then $$\mathrm{\mid }{D}_H{T}_{\mathrm{\Omega }}(t)f{\mathrm{\mid }}_H^p\le K_p{t}^{-\frac{p}{2}}{T}_{\mathrm{\Omega }}(t)\mathrm{\mid }f{\mathrm{\mid }}^p,t>0,\nu \text{-a.e. in }\mathrm{\Omega },$$ where $K_p$ is a positive constant depending only on $p$. Finally we investigate on the asymptotic behaviour of the semigroup $(T_{\mathrm{\Omega }}(t){)}_{t\ge 0}$ as $t$ goes to infinity.