## The Annals of Probability

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

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

#### Article information

Source
Ann. Probab., Volume 42, Number 4 (2014), 1297-1336.

Dates
First available in Project Euclid: 3 July 2014

https://projecteuclid.org/euclid.aop/1404394065

Digital Object Identifier
doi:10.1214/13-AOP853

Mathematical Reviews number (MathSciNet)
MR3262479

Zentralblatt MATH identifier
1318.60068

#### Citation

Cerrai, Sandra; Da Prato, Giuseppe. A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise. Ann. Probab. 42 (2014), no. 4, 1297--1336. doi:10.1214/13-AOP853. https://projecteuclid.org/euclid.aop/1404394065

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