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

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Cozzi 55, 20125 Milano, Italy.
² Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via per Arnesano, 73100 Lecce, Italy.
³ Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy.
gianmario.tessitore@unimib.it

Received: 16 June 2015
Accepted: 13 April 2016

Abstract

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