Existence and space-time regularity for stochastic heat equations on p.c.f. fractals

TL;DR

This paper establishes the existence and regularity of solutions to stochastic heat equations on p.c.f. fractals, showing that spectral dimension influences solution properties similarly to Euclidean spaces.

Contribution

It introduces a framework for solving stochastic heat equations on p.c.f. fractals and extends regularity results based on spectral dimension, generalizing classical Euclidean findings.

Findings

Existence of function-valued solutions for spectral dimension < 2

Hölder continuity of solutions in space and time

Spectral dimension governs regularity, not geometric dimension

Abstract

We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly H\"older continuous in space and time. We calculate the respective H\"older exponents, which extend the well-known results on the H\"older exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.