Probability Theory and Related Fields

, Volume 152, Issue 1–2, pp 265–297 | Cite as

Singular perturbations to semilinear stochastic heat equations

  • Martin Hairer


We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε = 0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities.

Mathematics Subject Classification (2000)

Primary 60H15 Secondary 60H30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Assing S.: A pregenerator for Burgers equation forced by conservative noise. Commun. Math. Phys. 225(3), 611–632 (2002)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ambrosio L., Savaré G., Zambotti L.: Existence and stability for Fokker-Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145(3–4), 517–564 (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Andres S., von Renesse M.-K.: Particle approximation of the Wasserstein diffusion. J. Funct. Anal. 258(11), 3879–3905 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Borell C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30(2), 207–216 (1975)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan T.: Scaling limits of Wick ordered KPZ equation. Commun. Math. Phys. 209(3), 671–690 (2000)MATHGoogle Scholar
  7. 7.
    De Giorgi E., Franzoni T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850 (1975)MATHMathSciNetGoogle Scholar
  8. 8.
    Da Prato G., Debussche A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Da Prato G., Debussche A., Tubaro L.: A modified Kardar-Parisi-Zhang model. Electron. Comm. Probab. 12, 442–453 (2007)MATHMathSciNetGoogle Scholar
  10. 10.
    Da Prato G., Kwapień S., Zabczyk J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23(1), 1–23 (1987)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Da Prato, G., Tubaro, L.: Wick powers in stochastic PDEs: an introduction. Preprint (2007)Google Scholar
  12. 12.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In: Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)Google Scholar
  13. 13.
    Fernique X.: Des résultats nouveaux sur les processus gaussiens. C. R. Acad. Sci. Paris Sér. A 278, 363–365 (1974)MATHMathSciNetGoogle Scholar
  14. 14.
    Freidlin M.: Some remarks on the Smoluchowski-Kramers approximation. J. Stat. Phys. 117(3–4), 617–634 (2004)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Hairer, M.: An introduction to stochastic PDEs. Lecture Notes. (2009)
  16. 16.
    Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. In: Probability and its Applications. A Modeling, White Noise Functional Approach. Birkhäuser Boston Inc., Boston MA (1996)Google Scholar
  17. 17.
    Hairer M., Stuart A.M., Voss J.: Analysis of SPDEs arising in path sampling. Part II: the nonlinear case. Ann. Appl. Probab. 17(5/6), 1657–1706 (2007)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Hairer, M., Stuart, A.M., Voss, J.: Sampling conditioned hypoelliptic diffusions. Ann. Appl. Probab. (2010, accepted)Google Scholar
  19. 19.
    Hairer M., Stuart A.M., Voss J., Wiberg P.: Analysis of SPDEs arising in path sampling. Part I: the Gaussian case. Commun. Math. Sci. 3(4), 587–603 (2005)MATHMathSciNetGoogle Scholar
  20. 20.
    Jona-Lasinio G., Mitter P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409–436 (1985)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Kolesnikov A.V.: Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures. J. Funct. Anal. 230(2), 382–418 (2006)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. In: Applications of Mathematics (New York), vol. 23. Springer-Verlag, Berlin (1992)Google Scholar
  23. 23.
    Kupferman R., Pavliotis G.A., Stuart A.M.: Itô versus Stratonovich white-noise limits for systems with inertia and colored multiplicative noise. Phys. Rev. E 70(3), 0361209 (2004)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Kuwae K., Shioya T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Commun. Anal. Geom. 11(4), 599–673 (2003)MATHMathSciNetGoogle Scholar
  25. 25.
    Mosco, U.: Approximation of the solutions of some variational inequalities. Ann. Scuola Norm. Sup. Pisa (3) 21, 373–394 (1967); erratum, ibid. (3) 21, 765 (1967)Google Scholar
  26. 26.
    Mosco U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Mikulevicius, R., Rozovskii, B.L.: On quantization of stochastic Navier-Stokes equation. Preprint (2010)Google Scholar
  28. 28.
    Pugachev O.V.: On Mosco convergence of diffusion dirichlet forms. Theory Probab. Appl. 53(2), 242–255 (2009)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1991)Google Scholar
  30. 30.
    Sudakov, V.N., Cirel’son, B.S.: Extremal properties of half-spaces for spherically invariant measures. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41, 14–24, 165 (1974). Problems in the theory of probability distributions, IIGoogle Scholar
  31. 31.
    Talagrand M.: Regularity of Gaussian processes. Acta Math. 159(1–2), 99–149 (1987)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Talagrand M.: Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81, 73–205 (1995)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Wong E., Zakai M.: On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematics DepartmentThe University of WarwickCoventryUK

Personalised recommendations