On a modelled rough heat equation

Abstract

We use the formalism of Hairer's regularity structures theory \cite{hai-14} to study a heat equation with non-linear perturbation driven by a space-time fractional noise. Different regimes are observed, depending on the global pathwise roughness of the noise. To this end, and following the procedure exhibited in \cite{hai-14}, the equation is first "lifted" into some abstract "regularity structure" and therein solved through a fixed-point argument. Then we construct a consistent "model" above the fractional noise, by relying on a smooth Fourier-type approximation of the process.