Elliptic boundary value problems with Gaussian white noise loads

Abstract

Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. The challenging cases of Neumann and Robin BVPs are considered.

The main obstacle for usual variational methods is the irregularity of the load. In particular, the Neumann boundary values are not well-defined.

In this work, the BVP is formulated by replacing the continuity of boundary trace mappings with measurability. Instead of variational methods alone, the novel BVP derives also from Cameron–Martin space techniques.

The new BVP returns the study of irregular white noise to the study of $L^2$-loads.