Viscosity and Minimax Solutions for Path-Dependent Hamilton–Jacobi Equations in Infinite Dimensions and Related Differential Games

Abstract

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 [2]. 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.