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

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.