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

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 des champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space W1,2(E;.), where . is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.