The Annals of Probability

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

Sandra Cerrai and Giuseppe Da Prato

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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


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References

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