Probability Theory and Related Fields

, Volume 135, Issue 3, pp 363–394 | Cite as

On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom

  • Sandra Cerrai
  • Mark Freidlin
Article
First Online:
Received:
Revised:

Abstract

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations μ u tt (t,x)=Δu(t,x)−u t (t,x)+b(x,u(t,x))+Q Open image in new window (t),u(0)=u 0, u t (0)=v 0, endowed with Dirichlet boundary conditions, converges as μ goes to zero to the solution of the semi-linear stochastic heat equation u t (t,x)=Δ u(t,x)+b(x,u(t,x))+Q Open image in new window (t),u(0)=u 0, endowed with Dirichlet boundary conditions. Moreover we consider relations between asymptotics for the heat and for the wave equation. More precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation, for any μ>0.

Key words or phrases

Smolukowski-Kramers approximation Stochastic semi-linear damped wave equations Stochastic semi-linear heat equations Stationary distributions Gradient systems Invariant measures 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and control of infinite-dimensional systems 1, Birkhäuser Boston, 1992Google Scholar
  2. 2.
    Dalang, R., Frangos, N.E.: The stochastic wave equation in two spatial dimensions. The Annals of Probability 26, 187–212 (1998)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Da Prato, G., Barbu, V.: The stochastic non-linear damped wave equation. Applied Mathematics and Optimization 46, 125–141 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge 1992Google Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. London Mathematical Society, Lecture Notes Series 229, Cambridge University Press, Cambridge 1996Google Scholar
  6. 6.
    Freidlin, M.I.: Random perturbations of reaction-diffusion equations: the quasi deterministic approximation. Transactions of the AMS 305, 665–697 (1988)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Freidlin, M.: Some remarks on the Smoluchowski-Kramers approximation. Journal of Statistical Physics 117, 617–634 (2004)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Gyöngy, I., Krylov, N.V.: Existence of strong solutions for Itô's stochastic equations via approximations. Probab. Theory Relat. Fields 103, 143–158 (1996)CrossRefGoogle Scholar
  9. 9.
    Karczewska, A., Zabczyk, J.: A note on stochastic wave equations. Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math. 215, Dekker (2001), pp. 501–511Google Scholar
  10. 10.
    Kramers, H.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Krylov, N.V.: Lectures on the Theory of Elliptic and Parabolic equations in Hölder Spaces. American Mathematical Society, Providence, RI, 1996Google Scholar
  12. 12.
    Lions, J.L.: Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. Lecture Notes in Mathematics 323, Springer Verlag, 1970Google Scholar
  13. 13.
    Millet, A., Sanz-Solé, M.: A stochastic wave equation in two space dimension: smoothness of the law. The Annals of Probability 27, 803–844 (1999)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Millet, A., Morien, P.: On a non-linear stochastic wave equation in the plane: existence and uniqueness of the solution. The Annals of Applied Probability 11, 922–951 (2001)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Ondreját, M.: Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process. Journal of Evolution Equations 4, 169–191 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Pavliotis, G.A., Stuart, A.: White noise limits for inertial particles in a random field. Multiscale Modeling and Simulation 1, 527–533 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Pavliotis, G.A., Stuart, A.: Periodic homogenization for hypoelliptic diffusions. Journal of Statistical Physics 117, 261–279 (2004)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, 1983Google Scholar
  19. 19.
    Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equation. Probab. Theory Relat. Fields 116, 421–443 (2000)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Peszat, S.: The Cauchy problem for a nonlinear stochastic wave equation in any dimension. Journal of Evolution Equations 2, 383–394 (2002)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Smoluchowski, M.: Drei Vortage über Diffusion Brownsche Bewegung und Koagulation von Kolloidteilchen. Physik Zeit. 17, 557–585 (1916)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sandra Cerrai
    • 1
  • Mark Freidlin
    • 2
  1. 1.Dip. di Matematica per le DecisioniUniversità di FirenzeFirenzeItaly
  2. 2.Department of MathematicsUniversity of MarylandMarylandUSA

Personalised recommendations