On an unbounded metric measure space of Ahlfors regularity H, whose L² space supports a suitable Dirichlet form with sub-Gaussian estimates for the kernel of the related semigroup, and where the associated diffusion has walk dimension W, one can explore the parabolic Anderson model: the heat equation with multiplicative Gaussian noise, where the noise is dependent on a parameter α. In this talk, we will define our notion of noise and solution and establish key lemmas to prove the existence of a solution for any α that satisfies 0 ≤ 2α < H / W < 1 + 2α. Finally, we will describe our proof of finite upper bounds for all moments, as well as the solution preserving positivity of our initial condition.
