A fractional Poisson equation: existence, regularity and approximations

TL;DR

This paper studies a stochastic boundary value problem involving fractional Brownian fields, establishing existence, regularity, and convergence of lattice approximations for the solution of the fractional Poisson equation.

Contribution

It introduces a new approach to solve fractional Poisson equations driven by fractional Brownian fields, proving existence, regularity, and convergence of approximations.

Findings

Existence and uniqueness of solutions established.

Sample path regularity proved.

Lattice approximation converges at a quantifiable rate.

Abstract

We consider a stochastic boundary value elliptic problem on a bounded domain $D\subset \mathbb{R}^k$, driven by a fractional Brownian field with Hurst parameter $H=(H_1,...,H_k)\in[{1/2},1[^k$. First we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove existence and uniqueness of solution, along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate.