We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbedby 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[exponents]1,2(E; μ), where μ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.

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

Da Prato, Giuseppe;Cerrai, Sandra
2014

Abstract

We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbedby 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[exponents]1,2(E; μ), where μ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.
2014
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/91988
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