Sobolev space theory and Hölder estimates for the stochastic partial differential equations on conic and polygonal domains

Abstract

We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equation $$du=\left(\sum _{i,j=1}^d{a}^{ij}{u}_{x^i{x}^j}+f^0+\sum _{i=1}^d{f}_{x^i}^i\right)dt+\sum _{k=1}^{\infty }{g}^{k}d{w}_t^k,t>0,x\in \mathcal{D}$$ given with non-zero initial data. Here $\{w_t^k:k=1,2,\cdots \}$ is a family of independent Wiener processes defined on a probability space $(\Omega ,\mathbb{P})$, $a^{ij}=a^{ij}(\omega ,t)$ are merely measurable functions on $\Omega \times (0,\infty )$, and $\mathcal{D}$ is either a polygonal domain in ${\mathbb{R}}^2$ or an arbitrary dimensional conic domain of the type \begin{equation} \label{conic} \mathcal{D}(\mathcal{M}):=\left\{x\in \mathbb{R}^d :\,\frac{x}{|x|}\in \mathcal{M}\right\}, \quad \quad \mathcal{M}\in S^{d-1}, \quad (d\geq 2) \end{equation} where $\mathcal{M}$ is an open subset of $S^{d-1}$ with $C^2$ boundary. We measure the Sobolev and Hölder regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp.