Laplace's equation, the nonlinear Poisson equation and the effects of Gaussian white noise on the boundary.

摘要

Elliptic partial differential equations (PDE's) and corresponding boundary value problems are well understood with a variety of boundary data. Over the past 25 years, an abundance of research has been done in stochastic PDE's (SPDE's), with an emphasis on equations having a time parameter on domains with low spatial dimension and whose boundary is smooth. The meaning of a solution to a class of elliptic SPDE's on a domain D ⊂ R d, d ≥ 2 with Lipschitz boundary ∂D is described. For this class of SPDE's, the randomness appears as a Gaussian white noise on the boundary of the domain. Existence, uniqueness and regularity results are obtained, and it is shown that these solutions are almost surely classical. For the Laplacian and the Helmholtz operator, the behavior of the solution near the boundary of the unit ball is described and in the case of the Laplacian, the solution is simply the harmonic extension of white noise and so many of the well-known properties of harmonic functions hold.