Regularity of the solutions to SPDEs in metric measure spaces

TL;DR

This paper investigates the regularity of solutions to nonlinear parabolic and stochastic PDEs on metric measure spaces with heat kernels, establishing Hölder continuity and applicability to fractional Brownian noise in low-dimensional fractal structures.

Contribution

It provides a novel analysis of regularity for PDEs and SPDEs on metric measure spaces, including fractional Brownian noise, using a-priori estimates in low spectral dimensions.

Findings

Solutions are Hölder continuous in time with values in fractional Sobolev spaces.

The approach applies to stochastic PDEs driven by fractional Brownian noise.

Works for metric measure spaces with spectral dimension less than 4.

Abstract

In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is H\"older continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4.