Stochastic Wave Equations Defined by Fractal Laplacians on Cantor-Like Sets

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 Hölder exponents. Moreover, we are concerned with the phenomenon of weak intermittency.