Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations

Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincaré inequalities with respect to an associated evolution system of measures $\{μ_t: t \in I\}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$.

Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R^d and in L^p-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in IxR^d, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in $\R^N$. We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator $\OU$, which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any $t>s$, the evolution operator $P_{s,t}$ is compact in the previous weighted spaces.

Log-Sobolev不等式与Ornstein-Uhlenbeck演化算子在无穷维空间中的超压缩性

Log-Sobolev inequalities and hypercontractivity for Ornstein-Uhlenbeck evolution operators in infinite dimensions

研究无穷维Hilbert空间中Ornstein-Uhlenbeck演化算子在$L^p$空间上的实现，通过Log-Sobolev估计证明超压缩性，并应用于非自治随机抛物PDE的转移演化算子。

Invariant Measures and Maximal L^2 Regularity for Nonautonomous Ornstein-Uhlenbeck Equations

We characterize the domain of the realization of the linear parabolic operator Gu := u_t + L(t)u (where, for each real t, L(t) is an Ornstein-Uhlenbeck operator), in L^2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L^2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators.

Nonautonomous Kolmogorov equations in the whole space: a survey on recent results

In this paper we survey some recent results concerned with nonautonomous Kolmogorov elliptic operators. Particular attention is paid to the case of the nonautonomous Ornstein-Uhlenbeck operator