Comparison of viscosity solutions for a class of non-linear PDEs on the space of finite nonnegative measures
Ibrahim Ekren, Xihao He, Tianxu Lan, Xiaolu Tan
TL;DR
The paper develops a viscosity-solution framework for nonlinear PDEs on the space of finite measures $\mathcal{M}_2(\mathbb{R}^d)$, extending existing results from the Wasserstein space to settings with mass nonconservation. It introduces a robust doubling-variables approach with a novel auxiliary function $\vartheta(t,m)=e^{-Lt}(\int\sqrt{1+|x|^2}m(dx)+m(\mathbb{R}^d)^2)$ to achieve compactness despite unbounded total mass, enabling a comparison principle for semicontinuous solutions. The authors apply this theory to a controlled McKean–Vlasov branching diffusion, deriving a priori estimates and the dynamic programming principle, and proving that the value function is the unique viscosity solution of the corresponding HJB equation. This yields a PDE-based approach to optimal control of branching processes and broadens the applicability of viscosity methods to measure-valued population dynamics with variable mass. The work thus provides both fundamental theoretical gains in the viscosity theory on $\mathcal{M}_2(\mathbb{R}^d)$ and practical tools for branching-control problems in biology, finance, and multi-agent systems.
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.