We study a full discretization scheme for the stochastic linear heat equation begin{equation*}begin{cases}partial_t langlePsirangle = Delta langlePsirangle +dot{B}, , quad tin [0,1], xin mathbb{R}, langlePsirangle_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 $langlePsirangle$, which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.
We construct a $K$-rough path above either a space-time or a spatial fractional Brownian motion, in any space dimension $d$. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood
in the renormalized sense. We also consider the case of a spatial fractional noise.
