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

We prove that a family of linear bounded evolution operators $({bf G}(t,s))_{tge sin I}$ 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 $Itimes Rd$ (where $I$ is a right-halfline or $I=R$) all having the same principal part. We establish some continuity and representation properties of $({bf G}(t,s))_{t ge sin I}$ 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.

Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $ItimesR^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincare inequalities with respect to an associated evolution system of measures ${mu_t: t in I}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$.