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July 2014 A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise
Sandra Cerrai, Giuseppe Da Prato
Ann. Probab. 42(4): 1297-1336 (July 2014). DOI: 10.1214/13-AOP853

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.

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Sandra Cerrai. Giuseppe Da Prato. "A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise." Ann. Probab. 42 (4) 1297 - 1336, July 2014. https://doi.org/10.1214/13-AOP853

Information

Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1318.60068
MathSciNet: MR3262479
Digital Object Identifier: 10.1214/13-AOP853

Subjects:
Primary: 35K57, 35R15, 60H15

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 4 • July 2014
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