The damped stochastic wave equation on p.c.f. fractals

TL;DR

This paper extends the stochastic wave equation to p.c.f. fractals, providing explicit solutions, analyzing their regularity, and exploring convergence properties, thus generalizing Euclidean results to fractal spaces with spectral dimension less than two.

Contribution

It introduces a novel formulation of the damped stochastic wave equation on p.c.f. fractals and derives explicit solutions and regularity properties, extending classical Euclidean results to fractal geometries.

Findings

Unique function-valued solutions exist with explicit spectral decomposition

Solutions exhibit specific H"older continuity in space and time

Convergence to equilibrium is established for the solutions

Abstract

A p.c.f. fractal with a regular harmonic structure admits an associated Dirichlet form, which is itself associated with a Laplacian. This Laplacian enables us to give an analogue of the damped stochastic wave equation on the fractal. We show that a unique function-valued solution exists, which has an explicit formulation in terms of the spectral decomposition of the Laplacian. We then use a Kolmogorov-type continuity theorem to derive the spatial and temporal H\"older exponents of the solution. Our results extend the analogous results on the stochastic wave equation in one-dimensional Euclidean space. It is known that no function-valued solution to the stochastic wave equation can exist in Euclidean dimension two or higher. The fractal spaces that we work with always have spectral dimension less than two, and show that this is the right analogue of dimension to express the "curse of dimensionality" of the stochastic wave equation. Finally we prove some results on the convergence to equilibrium of the solutions.