H\"{o}lder Continuity of the Solution for a Class of Nonlinear SPDE Arising from One Dimensional Superprocesses

TL;DR

This paper establishes sharp H"older continuity results for solutions to a class of nonlinear SPDEs from one-dimensional superprocesses, improving previous bounds using Malliavin calculus.

Contribution

It introduces a new approach employing Malliavin calculus to prove sharper H"older continuity exponents for nonlinear SPDE solutions.

Findings

Time H"older exponent close to 1/4, an improvement from 1/10

Spatial H"older exponent of 1/2 obtained

Results are sharp, matching linear heat equation properties

Abstract

The H\"older continuity of the solution to a nonlinear stochastic partial differential equation arising from one dimensional super process is obtained. It is proved that the H\"older exponent in time variable is as close as to 1/4, improving the result of 1/10 in a recent paper by Li et al [3]. The method is to use the Malliavin calculus. The H\"older continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This H\"older continuity result is sharp since the corresponding linear heat equation has the same H\"older continuity.