Stochastic Wave Equations defined by Fractal Laplacians on Cantor-like Sets

TL;DR

This paper investigates stochastic wave equations on fractal sets, providing new estimates, establishing solution existence, and analyzing regularity and intermittency phenomena.

Contribution

It introduces improved eigenfunction estimates and develops a framework for solving stochastic wave equations on Cantor-like fractal sets.

Findings

Established existence and uniqueness of solutions.

Proved H"older continuity in space and time.

Analyzed weak intermittency phenomena.

Abstract

We study stochastic wave equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we give an improved estimate on the uniform norm of eigenfunctions and approximate the wave propagator using the resolvent density. Afterwards, we establish existence and uniqueness of mild solutions to stochastic wave equations provided some Lipschitz and linear growth conditions. We prove H\"older continuity in space and time and compute the H\"older exponents. Moreover, we are concerned with the phenomenon of weak intermittency.