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.
Citation
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
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