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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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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Ann. Probab., Volume 42, Number 4 (2014), 1297-1336.

First available in Project Euclid: 3 July 2014

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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

Stochastic reaction–diffusion equations Kolmogorov operators Poincaré inequality spectral gap Sobolev spaces in infinite-dimensional spaces


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.

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  • [1] Bakry, D. and Émery, M. (1984). Hypercontractivité de semi-groupes de diffusion. C. R. Acad. Sci. Paris Sér. I Math. 299 775–778.
  • [2] Cerrai, S. (1994). A Hille–Yosida theorem for weakly continuous semigroups. Semigroup Forum 49 349–367.
  • [3] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension. A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, Berlin.
  • [4] Cerrai, S. (2003). Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304.
  • [5] Cerrai, S. (2005). Stabilization by noise for a class of stochastic reaction–diffusion equations. Probab. Theory Related Fields 133 190–214.
  • [6] Cerrai, S. (2006). Asymptotic behavior of systems of stochastic partial differential equations with multiplicative noise. In Stochastic Partial Differential Equations and Applications—VII. Lect. Notes Pure Appl. Math. 245 61–75. Chapman & Hall, Boca Raton, FL.
  • [7] Cerrai, S. (2011). Averaging principle for systems of reaction–diffusion equations with polynomial nonlinearities perturbed by multiplicative noise. SIAM J. Math. Anal. 43 2482–2518.
  • [8] Da Prato, G., Debussche, A. and Goldys, B. (2002). Some properties of invariant measures of non symmetric dissipative stochastic systems. Probab. Theory Related Fields 123 355–380.
  • [9] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [10] Donati-Martin, C. and Pardoux, É. (1993). White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 1–24.
  • [11] Priola, E. (1999). On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136 271–295.