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 identit\'e du carr\'e 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 Poincar\'e inequality and of the spectral gap.