Viscosity and minimax solutions for path-dependent Hamilton-Jacobi equations in infinite dimensions and related differential games

Abstract

We establish new results for path-dependent Hamilton-Jacobi equations with nonlinear monotone, and coercive operators on Hilbert space, which were initially studied in Bayraktar and Keller [J. Funct. Anal., 275 (8) (2018), pp. 2096-2161]. Under more general assumptions than in the cited paper (and more general than in the finite-dimensional case as well), we prove the uniqueness of a minimax solution of a terminal-value problem for the equation under consideration and the existence of such a solution on the whole path space. We introduce a new notion of a viscosity solution for this problem and show the equivalence of this notion to the notion of a minimax solution, which implies the corresponding existence and uniqueness theorem for viscosity solutions. In addition, we obtain a stability result for viscosity solutions using the half-relaxed limits method. As applications, we prove two theorems on the existence and characterization of value of a zero-sum differential game for a time-delay (path-dependent) evolution equation. The first theorem pertains to the case of non-anticipative (Elliott-Kalton) strategies and is related to the results on viscosity solutions, and the second theorem deals with the case of feedback (Krasovskii-Subbotin) strategies and is based on the results on minimax solutions.