A full discretization of the rough fractional linear heat equation

Abstract

We study a full discretization scheme for the stochastic linear heat equation

when $\dot{B}$ is a very rough space-time fractional noise.

The discretization procedure is divised into three steps: $(i)$ regularization of the noise through a mollifying-type approach; $(ii)$ discretization of the (smoothened) noise as a finite sum of Gaussian variables over rectangles in $[0,1]\times \mathbb{R}$; $(iii)$ discretization of the heat operator on the (non-compact) domain $[0,1]\times \mathbb{R}$, along the principles of Galerkin finite elements method.

We establish the convergence of the resulting approximation to , which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.