Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains

Abstract

We consider the stochastic heat equation ${\partial }_{t}u(t,x)=\Delta u(t,x)+{\dot{W}}_{\alpha }(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where ${\dot{W}}_{\alpha }$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta {)}^{-\alpha }$ with $\alpha >0$. We prove that a unique pointwise defined mild solution exists if and only if $\alpha >d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<\alpha <d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.

30 pages