Stochastic solutions of nonlinear PDE's and an extension of superprocesses

TL;DR

This paper introduces an extension of superprocesses to signed measures and distributions, broadening the applicability of stochastic solutions for nonlinear PDEs and enhancing parallel computation methods.

Contribution

It proposes a new class of superprocesses on signed measures and distributions, expanding stochastic solution techniques to more complex nonlinear PDEs.

Findings

Superprocesses on signed measures enable solutions for a wider class of PDEs.

The approach enhances parallel computation capabilities.

Provides a rigorous framework for stochastic solutions of nonlinear PDEs.

Abstract

Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: MacKean's and superprocesses. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE's. A new class of superprocesses, namely superprocesses on signed measures and on distributions, is proposed to extend the stochastic solution approach to a wider class of PDE's.