The Parabolic Anderson Model's Total Mass at Small Times: Geometry, Fluctuations, and Renormalization

Let $D\subset\mathbb R^d$ be a bounded domain, let $κ>0$ be fixed, and let $W$ be a fractional Brownian sheet on $\mathbb R\times\mathbb R^d$. Consider the Stratonovich parabolic Anderson model (PAM) $\partial_tu_κ=(\frac12Δ+κW')u_κ$ with Dirichlet boundary condition on $D$ and the flat initial condition $u_κ(0,\cdot)=\mathbf 1_D$. We calculate exact asymptotics for the expectation and the standard deviation of the total mass $\int_Du_κ(t,x)~\mathrm d x$ as $t\to0$ under the assumption that $W$'s Hurst indices are all at least $1/2$ and that $u_κ$'s moments are finite for small enough $t>0$. In doing so, we uncover that these asymptotics are determined by a competition between three mechanisms: (1) $\mathbf{Geometry}$: The rate of heat diffusion through the boundary $\partial D$. (2) $\mathbf{Fluctuations}$: $W$'s time Hurst index. (3) $\mathbf{Renormalization}$: The singularity of deterministic Stratonovich corrections. As a result, we identify novel phase transition phenomena, which arise from the influence of $W$'s Hurst indices on the relative magnitudes of these contributions.