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

2012/12/21 by Sandra Cerrai, Cerrai, Sandra, Giuseppe Da Prato +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1212.5376

Key words: Stochastic reaction-diffusion equations, Kolmogorov operators, Poincaré inequality, spectral gap, Sobolev spaces in infinite dimensional spaces

arxiv created 2012/12/21 · arxiv updated 2012/12/24

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 di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space W^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.

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