A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise

Abstract

We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type, in the Banach space $E$ of continuous functions. By analyzing the smoothing properties of the associated transition semigroup, we prove a modification of the classical identité du carré di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space $W^{1,2}(E;\mu )$, where $\mu$ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.

Key words: Stochastic reaction-diffusion equations, Kolmogorov operators, Poincaré inequality, spectral gap, Sobolev spaces in infinite dimensional spaces

Cited by in corpus (1)

• Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component