Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients

TL;DR

This paper investigates the properties of evolution operators linked to nonautonomous elliptic Kolmogorov operators with unbounded coefficients, focusing on their compactness and invariance in various function spaces.

Contribution

It establishes the existence of evolution operators for these Kolmogorov operators and analyzes their compactness and invariance properties in different functional settings.

Findings

Existence of evolution operators for nonautonomous elliptic operators with unbounded coefficients.

Conditions under which these operators are compact.

Criteria for invariance of operators in L^p and C_0 spaces.

Abstract

In this paper we consider nonautonomous elliptic operators ${\mathcal A}$ with nontrivial potential term defined in $I\times\mathbb R^d$, where $I$ is a right-halfline (possibly $I=\mathbb R$). We prove that we can associate an evolution operator $(G(t,s))$ with ${\mathcal A}$ in the space of all bounded and continuous functions on $\mathbb R^d$. We also study the compactness properties of the operator $G(t,s)$. Finally, we provide sufficient conditions guaranteeing that each operator $G(t,s)$ preserves the usual $L^p$-spaces and $C_0(\mathbb R^d)$.