## The Annals of Probability

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

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

Sandra Cerrai and Giuseppe Da Prato

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/13-AOP853

**Mathematical Reviews number (MathSciNet)**

MR3262479

**Zentralblatt MATH identifier**

1318.60068

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 35K57: Reaction-diffusion equations

**Keywords**

Stochastic reaction–diffusion equations Kolmogorov operators Poincaré inequality spectral gap Sobolev spaces in infinite-dimensional spaces

#### 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