Conservation laws of mean field games equations

TL;DR

This paper investigates conservation laws in mean field games equations with separable Hamiltonians, using symmetry analysis and variational principles to identify conserved quantities and symmetries.

Contribution

It introduces a systematic method to derive conservation laws for mean field games equations with separable Hamiltonians through symmetry analysis.

Findings

Identified symmetries of the mean field games system.

Derived conservation laws using Noether's theorem.

Found particular cases with additional symmetries and conservation laws.

Abstract

Mean field games equations are examined for conservation laws. The system of mean field games equations consists of two partial differential equations: the Hamilton-Jacobi-Bellman equation for the value function and the forward Kolmogorov equation for the probability density. For separable Hamiltonians, this system has a variational structure, i.e., the equations of the system are Euler-Lagrange equations for some Lagrangian functions. Therefore, one can use the Noether theorem to derive the conservation laws using variational and divergence symmetries. In order to find such symmetries, we find symmetries of the PDE system and select variational and divergence ones. The paper considers separable, state-independent Hamiltonians in one-dimensional state space. It examines the most general form of the mean field games system for symmetries and conservation laws and identifies particular cases of the system which lead to additional symmetries and conservation laws.