Short-time existence and uniqueness for some infinite-dimensional Nash systems
Davide Francesco Redaelli
TL;DR
The paper proves local-in-time existence and uniqueness for an infinite-dimensional Nash system of backward parabolic HJB equations on ${\mathbb{R}}^\omega_T$, motivated by stochastic games on sparse graphs. The key idea is to approximate the infinite system by finite-dimensional Nash systems and to establish a priori estimates that are stable with respect to dimension, achieved via a decaying, c-self controlled sequence $\beta$ with $\beta \star \beta \le c\beta$. A central technical contribution is a priori control for linear drift-diffusion equations (and their adjoints) in weighted Hölder spaces, yielding dimension-stable bounds on derivatives up to third order and Hölder continuity in time. Using a fixed-point argument in finite dimensions and a diagonalization/compactness pass to the limit, the authors obtain a unique classical solution in weighted spaces $u^i\in \ell^{\infty}(\mathbb{N}; C^0([0,T]; C^{2,1}_{\beta_i}(\mathbb{R}^\omega)) \cap C^{1/2}([0,T]; C^{2-}_{\sqrt{\beta_i}}(\mathbb{R}^\omega)))$ for small $T$, providing a rigorous PDE foundation for infinite-dimensional Nash systems arising from sparse-graph mean field game frameworks.
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.