A full discretization of the rough fractional linear heat equation

We study a full discretization scheme for the stochastic linear heat equation\begin{equation*}\begin{cases}\partial_t \=\Delta \+\dot{B}\, , \quad t\in [0,1], \ x\in \R,\\\_0=0\, ,\end{cases}\end{equation*}when $\dot{B}$ is a very \emph{rough space-time fractional noise}.\smallskipThe 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 $[0,1]\times \R$; $(iii)$ discretization of the heat operator on the (non-compact) domain $[0,1]\times \R$, along the principles of Galerkin finite elements method. \smallskipWe 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