Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains
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| Formato: | Preprint |
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2018
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| _version_ | 1866916896668909568 |
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| author | Angiuli, Luciana Ferrari, Simone Pallara, Diego |
| author_facet | Angiuli, Luciana Ferrari, Simone Pallara, Diego |
| contents | Let $X$ be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure $γ$ and let $λ_1$ be the maximum eigenvalue of the covariance operator associated with $γ$. The associated Cameron--Martin space is denoted by $H$. For a sufficiently regular convex function $U:X\to\mathbb{R}$ and a convex set $Ω\subseteq X$, we set $ν:=e^{-U}γ$ and we consider the semigroup $(T_Ω(t))_{t\geq 0}$ generated by the self-adjoint operator defined via the quadratic form \[ (φ,ψ)\mapsto \int_Ω\langle D_Hφ,D_Hψ\rangle_Hdν, \] where $φ,ψ$ belong to $D^{1,2}(Ω,ν)$, the Sobolev space defined as the domain of the closure in $L^2(Ω,ν)$ of $D_H$, the gradient operator along the directions of $H$.
A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_Ω(t))_{t\ge 0}$. 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 $f\in D^{1,p}(Ω,ν)$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in $Ω$ for the measure $ν$ and some improving summability properties for $(T_Ω(t))_{t\geq 0}$. In addition we prove that if $f$ belongs to $L^p(Ω,ν)$ 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 $(T_Ω(t))_{t\geq 0}$ as $t$ goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_07780 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains Angiuli, Luciana Ferrari, Simone Pallara, Diego Analysis of PDEs Let $X$ be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure $γ$ and let $λ_1$ be the maximum eigenvalue of the covariance operator associated with $γ$. The associated Cameron--Martin space is denoted by $H$. For a sufficiently regular convex function $U:X\to\mathbb{R}$ and a convex set $Ω\subseteq X$, we set $ν:=e^{-U}γ$ and we consider the semigroup $(T_Ω(t))_{t\geq 0}$ generated by the self-adjoint operator defined via the quadratic form \[ (φ,ψ)\mapsto \int_Ω\langle D_Hφ,D_Hψ\rangle_Hdν, \] where $φ,ψ$ belong to $D^{1,2}(Ω,ν)$, the Sobolev space defined as the domain of the closure in $L^2(Ω,ν)$ of $D_H$, the gradient operator along the directions of $H$. A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_Ω(t))_{t\ge 0}$. 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 $f\in D^{1,p}(Ω,ν)$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in $Ω$ for the measure $ν$ and some improving summability properties for $(T_Ω(t))_{t\geq 0}$. In addition we prove that if $f$ belongs to $L^p(Ω,ν)$ 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 $(T_Ω(t))_{t\geq 0}$ as $t$ goes to infinity. |
| title | Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1807.07780 |