On coupled systems of Kolmogorov equations with applications to stochastic differential games

TL;DR

This paper develops a theoretical framework for associating evolution operators with systems of elliptic operators having unbounded coefficients, providing continuity, representation, and compactness results, along with gradient estimates.

Contribution

It introduces a novel association between evolution operators and elliptic systems with unbounded coefficients, including new continuity, representation, and compactness properties.

Findings

Established a family of bounded evolution operators for elliptic systems.

Proved continuity and representation properties of these operators.

Derived a uniform weighted gradient estimate and its implications.

Abstract

We prove that a family of linear bounded evolution operators $({\bf G}(t,s))_{t\ge s\in I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $\bm{\mathcal A}$ with unbounded coefficients defined in $I\times \Rd$ (where $I$ is a right-halfline or $I=\R$) all having the same principal part. We establish some continuity and representation properties of $({\bf G}(t,s))_{t \ge s\in I}$ and a sufficient condition for the evolution operator to be compact in $C_b(\Rd;\R^m)$. We prove also a uniform weighted gradient estimate and some of its more relevant consequence.