On a modelled rough heat equation

Aurélien Deya (1)

1 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: Regularity structures theory; Stochastic PDEs; Rough paths theory; space-time Fractional Brownian motion

Domaines: Probabilités [math.PR]

Aurélien Deya. On a modelled rough heat equation. Probability Theory and Related Fields, 2016, 165 (1-2), pp.1-65. ⟨10.1007/s00440-015-0650-8⟩. ⟨hal-00981345v2⟩