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

TL;DR

This paper establishes a modified identity for Kolmogorov operators in the space of continuous functions related to reaction-diffusion equations with multiplicative noise, enabling the construction of Sobolev spaces and proving spectral gap properties.

Contribution

It introduces a new version of the carré du champs identity for non-Hilbertian settings, specifically for reaction-diffusion systems with polynomial growth and multiplicative noise.

Findings

Constructed the Sobolev space W^{1,2}(E;μ)

Proved the Poincaré inequality in this setting

Established the spectral gap for the invariant measure

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 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.