A fractional Poisson equation: existence, regularity and approximations

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 $H=(H_1,\mathrm{.}\mathrm{.}\mathrm{.},H_k)\in [1\mathrm{/}2,1{[}^k$. 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.