Stationary Mean Field Games on networks with sticky transition conditions

TL;DR

This paper investigates stochastic Mean Field Games on networks where agents can linger at vertices, analyzing the associated PDEs with generalized boundary conditions and the structure of invariant measures.

Contribution

It introduces a novel framework for Mean Field Games on networks with sticky transition conditions, extending PDE analysis to include generalized Kirchhoff conditions at vertices.

Findings

Characterization of the invariant measure as a combination of absolutely continuous and Dirac measures.

Derivation of Hamilton-Jacobi-Bellman equations with generalized Kirchhoff conditions.

Analysis of the generator's limitations concerning second-order derivatives.

Abstract

We study stochastic Mean Field Games on networks with sticky transition conditions. In this setting, the diffusion process governing the agent's dynamics can spend finite time both in the interior of the edges and at the vertices. The corresponding generator is subject to limitations concerning second-order derivatives and the invariant measure breaks down into a combination of an absolutely continuous measure within the edges and a sum of Dirac measures positioned at the vertices. Additionally, the value function, solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized Kirchhoff conditions at the vertices.