Superprocesses on ultradistributions

TL;DR

This paper introduces superprocesses on ultradistributions, expanding stochastic solution methods for nonlinear PDEs to handle a broader class of equations with diverse boundary conditions.

Contribution

It proposes a novel class of superprocesses on ultradistributions, overcoming limitations of traditional superprocesses restricted to measures.

Findings

Superprocesses on ultradistributions extend stochastic solutions to more nonlinear PDEs.

The approach handles arbitrary boundary conditions.

Provides a new framework for probabilistic solutions of complex 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: McKean's and superprocesses. In favour of superprocesses is the fact that they handle arbitrary boundary conditions. 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 ultradistributions, is proposed to extend the stochastic solution approach to a wider class of PDE's.