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

Abstract

We study stochastic heat equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we first investigate the corresponding heat kernels. Then, we prove existence and uniqueness of mild solutions to stochastic heat equations provided some Lipschitz and linear growth conditions. We establish H\"older continuity in space and time and compute the H\"older exponents. Moreover, we address the question of weak intermittency.