Dirichlet problem for stochasticparabolic equations in smoothdomains
Sergey V. Lototsky · Stochastics and stochastics reports · 1999
A second-order stochastic parabolic equation with zero Dirichlet boundary conditions is considered in a sufficiently smooth bounded domain. Existence, uniqueness, and regularity of the solution are established without assuming any compatibility relations. To control the solution near the boundary of the region, special Sobolev-type spaces with weights are introduced. To illustrate the results, two examples are considered: general linear equation with finite-dimensional noise and equation on a line segment, driven by space-time white noise