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 di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space $W^{1,2}(E;μ)$, where $μ$ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.

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