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

Addona, Davide; Angiuli, Luciana; Lorenzi, Luca; Tessitore, Gianmario

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 937-976.

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

Classification : 35K45, 35K58, 47B07, 60H10, 91A15
Mots clés : Nonautonomous parabolic systems, unbounded coefficients, evolution operators, compactness, gradient estimates, semilinear systems, stochastic games

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