Short-time existence and uniqueness for some infinite-dimensional Nash systems

TL;DR

This paper establishes local existence and uniqueness results for a class of infinite-dimensional Nash systems, which are systems of Hamilton-Jacobi-Bellman equations in infinite-dimensional spaces, with implications for stochastic differential games.

Contribution

It introduces new local well-posedness results for infinite-dimensional Nash systems and provides dimension-stable a priori estimates for related transport-diffusion equations.

Findings

Proved local existence and uniqueness for infinite-dimensional Nash systems.

Established dimension-stable a priori estimates for transport-diffusion equations.

Connected Nash systems to stochastic differential games with sparse graph interactions.

Abstract

We prove local (in time) existence and uniqueness for a class of infinite-dimensional Nash systems, namely systems of infinitely many Hamilton-Jacobi-Bellman equations set in an infinite-dimensional Euclidean space. Such systems have been recently showed (see arXiv:2401.06534) to arise in the theory of stochastic differential games with interactions governed by sparse graphs, under structural assumptions that inspired the hypotheses exploited in the present work. Contextually, we also prove a general linear result, providing a priori estimates, stable with respect to the dimension, for transport-diffusion equations whose drifts (and their derivatives) enjoy appropriate decay properties.