Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 653–684 | Cite as

A probabilistic Harnack inequality and strict positivity of stochastic partial differential equations

  • Zhenan Wang


Under general conditions we show an a priori probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form
$$\begin{aligned} \partial _t u=\mathrm {div}{\; (}{\mathbb {A}}\nabla u)+f(t,x,u;\omega )+g_i(t,x,u;\omega )\dot{w}_t^i. \end{aligned}$$
We also show that the solutions of the above equation are almost surely strictly positive if the initial condition is non-negative and not identically vanishing.

Mathematics Subject Classification




The project was initially started by the author and Doctor Yu Wang (currently in Goldman Sachs) in 2014 upon the completion of [6]. Although the collaboration ended after the departure of Yu Wang, the discussion with him has helped clarify many confusions. His contribution to this project is greatly appreciated.


  1. 1.
    Carmona, R., Rozovskii, B.L. (eds.): Stochastic Partial Differential Equations: Six Perspectives. Mathematical Surveys and Monographs Series, vol. 64. American Mathematical Society, Providence (1998)Google Scholar
  2. 2.
    Dareiotis, K., Gerencsér, M.: Local \(L_\infty \)-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs. J. Differ. Equ. 262(1), 615C632 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Debussche, A., De Moor, S., Hofmanova, M.: A Regularity Result for Quasilinear Stochastic Partial Differential Equations of Parabolic Type. arXiv:1401.6369
  4. 4.
    Fabes, E.B., Garofalo, N.: Parabolic BMO and Harnack’s inequality. In: Proceedings of the American Mathematical Society, pp. 63–69 (1985)Google Scholar
  5. 5.
    Han, Q., Lin, F.: Elliptic Partial Differential Equations, vol. 1. American Mathematical Society (2011)Google Scholar
  6. 6.
    Hsu, E.P., Wang, Y., Wang, Z.: Stochastic De Giorgi Iteration and Regularity of Stochastic Partial Differential Equation. arXiv:1312.3311. To appear in Annals of Probability
  7. 7.
    Krylov, N.V.: An analytic approach to SPDEs. In: Stochastic Partial Differential Equations: Six Perspectives. Mathematical Surveys and Monographs, vol. 64, pp. 185-242. AMS, Providence (1999)Google Scholar
  8. 8.
    Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17(1), 101–134 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mueller, C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pardoux, É.: Equations aux Derivées Partielles Stochastiques Monotones, Thèse, Univ. Paris-Sud (1975)Google Scholar
  11. 11.
    Pardoux, É.: Stochastic Partial Differential Equations. Lecture notes for the course given at Fudan University (2007)Google Scholar
  12. 12.
    Tessitore, G., Zabczyk, J.: Strict positivity for stochastic heat equations. Stoch. Process. Appl. 77(1), 83–98 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University of WashingtonSeattleUSA

Personalised recommendations