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

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.2000 Mathematics Subject Classification. 35K45; 35K58, 47B07, 60H10, 91A15.Remark 2.4. (i) Hypothesis 2.2(i) can be replaced with the weaker requirement that K η,ε is bounded from below in J × R d , uniformly with respect to η ∈ ∂B 1 , for any bounded interval J ⊂ I. Indeed, in this case, for any J as above, let c J > 0 be such that K η,ε ≥ −c J in J × R d for any η ∈ ∂B 1 . The change of unknowns v(t, x) := e −cJ (t−s)/4 u(t, x) transforms the elliptic operator A into the operator A − c J /4, which satisfies Hypothesis 2.2(i) and, clearly, the uniqueness of v is equivalent to the uniqueness of u. (ii) In the scalar case when the elliptic operator in (1.1) is A = Tr(QD 2 ) + b, ∇ + c and c is bounded from above (otherwise, Proposition 2.5 fails in general), taking ε = 1 and κ = c, one easily realizes that Hypothesis 2.2(i) is trivially satisfied. Moreover, Hypothesis (2.2)(ii) reduces to require the existence of a Lyapunov function for the operator A+ c, for any bounded interval J ⊂ I. This condition seems to be much more general than that typically assumed

‐Theory for Schrödinger systems

“…This shows that the unit ball of ( + ) is covered by the balls in ( R ; C ) centered at of radius 2 1 . As > 0 was arbitrary, it follows that the unit ball of ( + ) is totally bounded in ( R ; C ) .…”

“…For antisymmetric potentials , a linear growth of the entries is possible. In this article, we will allow potential terms whose entries grow like | | for some ∈ [1,2). Note that the results obtained here do not follow from [18] by setting = 0.…”

“…Besides being interesting in their own right, such systems appear naturally in the study of backward‐forward stochastic differential systems, in the study of Nash equilibria to stochastic differential games, in the analysis of the weighted ‐problem in , in the time‐dependent Born–Openheimer theory and also in the study of Navier–Stokes equations. We refer the reader to [, Section 6], for further details. In these applications unbounded drift and diffusion coefficients as well as unbounded potential terms appear.…”

“…The strategy in these references is quite different from that in . Namely, in solutions to the parabolic equation are at first constructed in the space of bounded and continuous functions. Afterwards the semigroup is extrapolated to the ‐scale.…”

“…Note that the results obtained here do not follow from [18] by setting = 0. Following [18], there were some other publications [1,2,11] with less restrictive assumptions on the coefficients; in particular, also unbounded diffusion coefficients were considered. The strategy in these references is quite different from that in [18].…”