Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses

  • 279 Accesses

  • 2 Citations

Abstract

The Hölder continuity of the solution X t (x) to a nonlinear stochastic partial differential equation (see (1.2) below) arising from one dimensional superprocesses is obtained. It is proved that the Hölder exponent in time variable is arbitrarily close to 1/4, improving the result of 1/10 in Li et al. (to appear on Probab. Theory Relat. Fields.). The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.

This is a preview of subscription content, log in to check access.

References

  1. 1

    Dalang R.C., Khoshnevisan D., Nualart E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3, 231–271 (2007)

  2. 2

    Dawson D.A., Li Z., Wang H.: Superprocesses with dependent spatial motion and general branching densities. Electronic J. Probab. 6, 1–33 (2001)

  3. 3

    Dawson D.A., Vaillancourt J., Wang H.: Stochastic partial differential equations for a class of interacting measure-valued diffusions. Ann. Inst. Henri. Poincaré Probab. Stat. 36, 167–180 (2000)

  4. 4

    Li, Z., Wang, H., Xiong, J., Zhou, X.: Joint continuity for the solutions to a class of nonlinear SPDE. Probab. Theory Relat. Fields (to appear)

  5. 5

    Konno N., Shiga T.: Stochasitc partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79, 201–225 (1988)

  6. 6

    Krylov, N.V.: An analytic approach to SPDEs, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr., 64, 185-242, Amer. Math. Soc., Providence, RI (1999)

  7. 7

    Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge (1990)

  8. 8

    Nualart, D.: The Malliavin calculus and related topics, 2nd edition. Springer (2006)

  9. 9

    Reimers M.: One-dimensional stochastic partial differerntial equations and the branching measure diffusion. Probab. Theory Related Fields 81, 319–340 (1989)

  10. 10

    Wang H.: State classification for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Related Fields 109, 39–55 (1997)

  11. 11

    Wang H.: A class of measure-valued branching diffusions in a random medium. Stoch. Anal. Appl. 16, 753–786 (1998)

Download references

Author information

Correspondence to Fei Lu.

Additional information

Yaozhong Hu is partially supported by a grant from the Simons Foundation #209206 and David Nualart is supported by the NSF grant DMS0904538.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hu, Y., Lu, F. & Nualart, D. Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses. Probab. Theory Relat. Fields 156, 27–49 (2013). https://doi.org/10.1007/s00440-012-0419-2

Download citation

Keywords

  • Nonlinear stochastic partial differential equation
  • Stochastic heat kernel
  • Conditional transition probability density in a random environment
  • Malliavin calculus
  • Hölder continuity
  • Moment estimates

Mathematics Subject Classification

  • 60H07
  • 60H15
  • 60H30