The Annals of Applied Probability

A Khasminskii type averaging principle for stochastic reaction–diffusion equations

Sandra Cerrai

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We prove that an averaging principle holds for a general class of stochastic reaction–diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite-dimensional systems.

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Ann. Appl. Probab., Volume 19, Number 3 (2009), 899-948.

First available in Project Euclid: 15 June 2009

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Primary: 60H15: Stochastic partial differential equations [See also 35R60] 34C29: Averaging method 37L40: Invariant measures

Stochastic reaction diffusion equations invariant measures ergodic and strongly mixing processes averaging principle


Cerrai, Sandra. A Khasminskii type averaging principle for stochastic reaction–diffusion equations. Ann. Appl. Probab. 19 (2009), no. 3, 899--948. doi:10.1214/08-AAP560.

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  • [1] Arnold, V. I., Kozlov, V. V. and Neishtadt, A. I. (2006). Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed. Encyclopaedia of Mathematical Sciences 3. Springer, Berlin.
  • [2] Bakhtin, V. and Kifer, Y. (2004). Diffusion approximation for slow motion in fully coupled averaging. Probab. Theory Related Fields 129 157–181.
  • [3] Bogoliubov, N. N. and Mitropolsky, Y. A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations. Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York.
  • [4] Brin, M. and Freidlin, M. (2000). On stochastic behavior of perturbed Hamiltonian systems. Ergodic Theory Dynam. Systems 20 55–76.
  • [5] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, Berlin.
  • [6] Cerrai, S. (2003). Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304.
  • [7] Cerrai, S. (2006). Asymptotic behavior of systems of stochastic partial differential equations with multiplicative noise. In Stochastic Partial Differential Equations and Applications—VII. Lecture Notes in Pure and Applied Mathematics 245 61–75. Chapman and Hall/CRC, Boca Raton, FL.
  • [8] Cerrai, S. and Freidlin, M. (2008). Averaging principle for a class of SPDE’s. In Probability Theory and Related Fields. To appear.
  • [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] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
  • [11] Fontbona, J. (2003). Nonlinear martingale problems involving singular integrals. J. Funct. Anal. 200 198–236.
  • [12] Freidlin, M. I. (2001). On stable oscillations and equilibriums induced by small noise. J. Statist. Phys. 103 283–300.
  • [13] Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, New York.
  • [14] Freidlin, M. I. and Wentzell, A. D. (2006). Long-time behavior of weakly coupled oscillators. J. Statist. Phys. 123 1311–1337.
  • [15] Gyöngy, I. and Krylov, N. (1996). Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields 103 143–158.
  • [16] Khasminskii, R. Z. (1968). On the principle of averaging the Itô’s stochastic differential equations. Kybernetika (Prague) 4 260–279.
  • [17] Kifer, Y. (2004). Some recent advances in averaging. In Modern Dynamical Systems and Applications 385–403. Cambridge Univ. Press, Cambridge.
  • [18] Kifer, Y. (2001). Averaging and climate models. In Stochastic Climate Models (Chorin, 1999). Progr. Probab. 49 171–188. Birkhäuser, Basel.
  • [19] Kifer, Y. (2001). Stochastic versions of Anosov’s and Neistadt’s theorems on averaging. Stoch. Dyn. 1 1–21.
  • [20] Kuksin, S. B. and Piatnitski, A. L. (2008). Khasminskii–Whitham averaging for randomly perturbed KdV equation. J. Math. Pures Appl. (9) 89 400–428.
  • [21] Maslowski, B., Seidler, J. and Vrkoč, I. (1991). An averaging principle for stochastic evolution equations. II. Math. Bohem. 116 191–224.
  • [22] Millet, A. and Morien, P.-L. (2001). On a nonlinear stochastic wave equation in the plane: existence and uniqueness of the solution. Ann. Appl. Probab. 11 922–951.
  • [23] Neishtadt, A. I. (1976). Averaging in multyfrequency systems. Soviet Physics Doktagy 21 80–82.
  • [24] Papanicolaou, G. C., Stroock, D. and Varadhan, S. R. S. (1977). Martingale approach to some limit theorems. In Papers from the Duke Turbulence Conference (Duke Univ., Durham, NC, 1976), Paper No. 6 Duke Mathematical Journal 3 120. Duke Univ. Press, Durham, NC.
  • [25] Seidler, J. and Vrkoč, I. (1990). An averaging principle for stochastic evolution equations. Časopis Pěst. Mat. 115 240–263.
  • [26] Tessitore, G. and Zabczyk, J. (2006). Wong–Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6 621–655.
  • [27] Veretennikov, A. Y. (1991). On an averaging principle for systems of stochastic differential equations. Mat. Sb. 69 271–284.
  • [28] Volosov, V. M. (1962). Averaging in systems of ordinary differential equations. Russian Math. Surveys 17 1–126.