Probabilistic Solution of the Dirichlet Problem for Biharmonic Functions in Discrete Space

Abstract

Considering difference equations in discrete space instead of differential equations in Euclidean space, we investigate a probabilistic formula for the solution of the Dirichlet problem for biharmonic functions. This formula involves the expectation of a weighted sum of the pay-offs at the successive times at which the Markov chain is in the complement of the domain. To make the infinite sum converge, we use Borel's summability method. This is interpreted probabilistically by imbedding the Markov chain into a continuous time, discrete space Markov process.