A generalization of the Hopf-Cole transformation for stationary Mean Field Games systems

TL;DR

This paper introduces a transformation that simplifies stationary Mean Field Games systems with superlinear Hamiltonians, converting the Hamilton-Jacobi-Bellman equation into a quasi-linear form involving the r-Laplace operator, under certain solution assumptions.

Contribution

It generalizes the Hopf-Cole transformation to a broader class of Mean Field Games systems with superlinear Hamiltonians.

Findings

Decouples stationary MFG systems with |p|^r Hamiltonians.

Transforms the HJB equation into an r-Laplace-based quasi-linear equation.

Applicable in 1D or for radial solutions.

Abstract

In this note we propose a transformation which decouples stationary Mean Field Games systems with superlinear Hamiltonians of the form |p|^r, and turns the Hamilton-Jacobi-Bellman equation into a quasi-linear equation involving the r-Laplace operator. Such a transformation requires an assumption on solutions of the system, which is satisfied for example in space dimension one or if solutions are radial.