Existence of densities for the dynamic $\Phi^4_3$ model

Paul Gassiat (1) , Cyril Labbé (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Université Paris Dauphine-PSL

We apply Malliavin calculus to the $\Phi^4_3$ equation on the torus and prove existence of densities for the solution of the equation evaluated at regular enough test functions. We work in the framework of regularity structures and rely on Besov-type spaces of modelled distributions in order to prove Malliavin differentiability of the solution. Our result applies to a large family of Gaussian space-time noises including white noise, in particular the noise may be degenerate as long as it is sufficiently rough on small scales.

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

Paul Gassiat, Cyril Labbé. Existence of densities for the dynamic $\Phi^4_3$ model. 2018. ⟨hal-01936380v1⟩