Comparison of viscosity solutions for a class of non-linear PDEs on the space of finite nonnegative measures

TL;DR

This paper develops a comparison principle for viscosity solutions of nonlinear PDEs on the space of finite measures, extending existing results and applying them to controlled branching processes.

Contribution

It introduces a comparison principle for PDEs on measure spaces and applies it to characterize the value function of controlled branching McKean-Vlasov diffusions.

Findings

Established a comparison principle for viscosity solutions on measure spaces.

Characterized the value function as the unique viscosity solution of a HJB equation.

Extended PDE methods to the control of branching processes.

Abstract

We establish a comparison principle for viscosity solutions of a class of nonlinear partial differential equations posed on the space of nonnegative finite measures, thereby extending recent results for PDEs defined on the Wasserstein space of probability measures. As an application, we study a controlled branching McKean-Vlasov diffusion and characterize the associated value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. This yields a PDE-based approach to the optimal control of branching processes.