Existence of the Fomin derivative of the invariant measure of a stochastic reaction--diffusion equation

Giuseppe da Prato (1) , Arnaud Debussche (2, 3)

1 SNS - Scuola Normale Superiore di Pisa
2 IPSO - Invariant Preserving SOlvers
3 IRMAR - Institut de Recherche Mathématique de Rennes

We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove existence of the Fomin derivative of the corresponding transition semigroup $P_t$. The main tool is a new estimate for $P_tD\varphi$ in terms of $\|\varphi\|_{L^2(H,\nu)}$, where $\nu$ is the invariant measure of $P_t$.

Mots clés: Stochastic reaction diffusion equations; invariant measures; Fomin differentiability

Domaines: Probabilités [math.PR]

Giuseppe da Prato, Arnaud Debussche. Existence of the Fomin derivative of the invariant measure of a stochastic reaction--diffusion equation. RIMS Workshop on Mathematical analysis of viscous incompressible fluid, Kyoto University, Nov 2014, Kyoto, Japan. ⟨hal-01126707⟩