Heat semigroup and singular PDEs

Ismaël Bailleul (1) , Frederic Bernicot (2, 3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 Equations aux dérivées partielles

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a coloured noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds. This machinery can be extended to an even more singular setting and deal with Sobolev spaces rather than Hölder spaces.

Mots clés: paracontrolled calculus; stochastic partial differential equations; heat semigroup; parabolic Anderson model equation

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Ismaël Bailleul, Frederic Bernicot. Heat semigroup and singular PDEs: With an Appendix by F. Bernicot and D. Frey. Journal of Functional Analysis, 2016, 270 (9), pp.3344-3452. ⟨hal-01110466v2⟩