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 perturbedby 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 W[exponents]1,2(E; μ), where μ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.