Stability analysis of a branching diffusion solver for semilinear heat equations

TL;DR

This paper investigates the stability of a stochastic branching method for solving semilinear heat equations, providing criteria for integrability and ensuring solution uniqueness in high-dimensional PDE contexts.

Contribution

It introduces new stability criteria for a branching diffusion solver, ensuring non-explosion and uniqueness of solutions under specific integrability conditions.

Findings

Derived sufficient criteria for integrability of branching processes.

Proved uniqueness of mild solutions under uniform integrability.

Enhanced understanding of stability conditions for stochastic PDE solvers.

Abstract

Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the integrability of random functionals of branching processes in order to ensure the non-explosion of solutions. In this paper, we study the stability of a functional branching representation of PDE solutions by deriving sufficient criteria for the integrability of the multiplicative weighted progeny of stochastic branching processes. We also prove the uniqueness of mild solutions under uniform integrability assumptions on random functionals.