A full discretization of the rough fractional linear heat equation

TL;DR

This paper develops a comprehensive discretization scheme for the rough fractional linear heat equation driven by very irregular space-time fractional noise, proving convergence in a distributional sense and illustrating the method with simulations.

Contribution

It introduces a novel three-step discretization approach combining noise regularization, finite Gaussian sum approximation, and Galerkin finite elements for a rough stochastic PDE.

Findings

Proves convergence of the discretization to the solution in a distribution space.

Provides a practical algorithm with partial simulations.

Addresses the challenge of discretizing equations driven by very rough noise.

Abstract

We study a full discretization scheme for the stochastic linear heat equation \begin{equation*}\begin{cases}\partial_t \langle\Psi\rangle = \Delta \langle\Psi\rangle +\dot{B}\, , \quad t\in [0,1], \ x\in \mathbb{R},\\ \langle\Psi\rangle_0=0\, ,\end{cases}\end{equation*} when $\dot{B}$ is a very \emph{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 $\langle\Psi\rangle$, which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.