Master equations with an individual noise on finite state graphs

TL;DR

This paper develops a rigorous mathematical framework for mean field games and related equations on finite graphs, incorporating individual noise and optimal transport structures, with applications to Nash equilibria.

Contribution

It introduces a well-posedness and regularity theory for master equations on finite graphs with individual noise, extending optimal transport and game theory models.

Findings

Established classical solutions for master equations on graphs.

Proved positivity preservation preventing boundary degeneracy.

Connected the system to Markov chain Nash equilibria.

Abstract

We develop a classical well-posedness and regularity theory on a finite connected weighted graph for an extended mean field game system, its associated master equation, and a Hamilton-Jacobi- Bellman equation on the probability simplex, all in the presence of an individual noise operator. The geometric structure is inherited from the logarithmic-mean activation functional of discrete optimal transport, under which the entropic Fokker-Planck equation appears as a gradient flow on the graph and the individual noise operator is a bilinear form in the probability vector and the Wasserstein gradient. A central technical step is a quantitative preservation-of-positivity estimate for the discrete continuity equation, which rules out finite-time boundary degeneracy and yields a classical solution theory for the master equation on the open simplex without imposing any boundary condition. As an application, we recover a Nash equilibrium interpretation of the discrete system in terms of Markov chains on the graph. Our setup is inspired by the computational algorithms for optimal mass transport of [10, 11] and provides a rigorous well-posedness theory for several of the equations derived in [25].