Elliptic boundary values problems with Gaussian white noise

Abstract

This work focuses on the study of elliptic boundary value problems with Gaussian white noise on the form LX = W, where L = - Δ + v², with v > 0. First, the definitions and some properties of some important tools for this study such as, Gaussian random field and Gaussian white noise are given. Next, the boundary value problem is solved in dimensions d = 1; 2, for the case of Dirichlet boundary condition. The spectral theorem is used to show that the operator L admits eigenfunctions that form a complete orthonormal basis and the boundary value problem is solved in the eigenbasis of L. Finally, the Bayesian estimation approach for inverse problem is derived, and as application, a one dimensional deconvolution problem is considered. The mathematical model is discretized on finite interval and the Bayesian estimation approach is used to estimate the truncated unknown.