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

Abstract

We prove that a family of linear bounded evolution operators $(\mathbf{G}(t,s){)}_{t\ge s\in I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $\mathcal{A}$ with unbounded coefficients defined in $I\times \Rd$ (where $I$ is a right-halfline or $I=\mathbb{R}$) all having the same principal part. We establish some continuity and representation properties of $(\mathbf{G}(t,s){)}_{t\ge s\in I}$ 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.