A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise. (English) Zbl 1318.60068

The authors study the reaction-diffusion equation \[ \partial_t u(t,\xi)= \partial^2_{\xi}u(t,\xi) + f(\xi,u)+g(\xi,u)\partial_t w(t,\xi),\quad t\geq0,\;\xi\in[0,1], \] perturbed by a multiplicative space-time white noise \(\partial_t w(t,\xi)\) and appropriate boundary conditions \(u(t,0)=u(t,1)=0\), \(u(0,\xi)=x(\xi)\). The nonlinear term \(f\) is assumed to be of polynomial growth in the second variable. Analyzing the associated Kolmogorov operator and transition semigroup, they prove a modification of the identity known as identité du carré des champs in the classical additive noise case. As an application of this identity, they construct the Sobolev space \(W^{1,2}(E,\mu)\), where \(E=C_0([0,1])\) and \(\mu\) is an invariant measure for the system, and they prove the validity of the Poincaré inequality and of the spectral gap.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35K57 Reaction-diffusion equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI arXiv


[1] Bakry, D. and Émery, M. (1984). Hypercontractivité de semi-groupes de diffusion. C. R. Acad. Sci. Paris Sér. I Math. 299 775-778. · Zbl 0563.60068
[2] Cerrai, S. (1994). A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum 49 349-367. · Zbl 0817.47048
[3] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension. A Probabilistic Approach. Lecture Notes in Math. 1762 . Springer, Berlin. · Zbl 0983.60004
[4] Cerrai, S. (2003). Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271-304. · Zbl 1027.60064
[5] Cerrai, S. (2005). Stabilization by noise for a class of stochastic reaction-diffusion equations. Probab. Theory Related Fields 133 190-214. · Zbl 1077.60046
[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. · Zbl 1093.60035
[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. · Zbl 1239.60055
[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. · Zbl 1087.60049
[9] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 . Cambridge Univ. Press, Cambridge. · Zbl 0761.60052
[10] Donati-Martin, C. and Pardoux, É. (1993). White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 1-24. · Zbl 0794.60059
[11] Priola, E. (1999). On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136 271-295. · Zbl 0955.47024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.