Singular Behavior of the Solution to the Stochastic Heat Equation on a Polygonal Domain

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We study the stochastic heat equation with trace class noise and zero Dirichlet boundary condition on a bounded polygonal domain O in R². It is shown that the solution u can be decomposed into a regular part uᵣ and a singular part uₛ which incorporates the corner singularity functions for the Poisson problem. Due to the temporal irregularity of the noise, both uᵣ and uₛ have negative L₂-Sobolev regularity of order s<-1/2 in time. The regular part uᵣ admits spatial Sobolev regularity of order r=2, while the spatial Sobolev regularity of uₛ is restricted by r<1+π/γ, where γis the largest interior angle at the boundary of O. We obtain estimates for the Sobolev norm of uᵣ and the Sobolev norms of the coefficients of the singularity functions. The proof is based on a Laplace transform argument w.r.t. the time variable. The result is of interest in the context of numerical methods for stochastic PDEs.