On the gradient estimates for evolution operators associated to Kolmogorov operators

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated to nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover we exhibit a class of operators which satisfy our conditions

On improvement of summability properties in nonautonomous Kolmogorov equations

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided

Non autonomous parabolic problems with unbounded coefficients in unbounded domains

Given a class of nonautonomous elliptic operators \A(t) with unbounded coefficients, defined in \overline{I \times \Om} (where $I$ is a right-halfline or $I=\R$ and \Om\subset \Rd is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to \A(t) in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved

LpL^p-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf G}(t,s)$ in $C_b({\mathbb R}^d;{\mathbb R}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb R}^d;{\mathbb R}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates

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 $\Omega\subseteq X$, we set $\nu:=e^{-U}\gamma$ and we consider the semigroup $(T_\Omega(t))_{t\geq 0}$ generated by the self-adjoint operator defined via the quadratic form $$(\varphi,\psi)\mapsto \int_\Omega\langle D_H\varphi,D_H\psi\rangle_Hd\nu,$$ 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 $$|D_H T_\Omega(t)f|_H^p\le e^{- p \lambda_1^{-1} t}(T_\Omega(t)|D_H f|^p_H), \qquad\, t>0,\ \nu\textrm{ -a.e. in }\Omega,$$ 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 Poincar\'e inequality in $\Omega$ for the measure $\nu$ and some improving summability properties for $(T_\Omega(t))_{t\geq 0}$. In addition we prove that if $f$ belongs to $L^p(\Omega,\nu)$ for some $p\in(1,\infty)$, then $$|D_H T_\Omega(t)f|^p_H \leq K_p t^{-\frac{p}{2}} T_\Omega(t)|f|^p,\qquad \, t>0,\ \nu\text{-a.e. in }\Omega,$$ where $K_p$ is a positive constant depending only on $p$. Finally we investigate on the asymptotic behaviour of the semigroup $(T_\Omega(t))_{t\geq 0}$ as $t$ goes to infinity

Invariant measures for systems of Kolmogorov equations

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity