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.
Article PDF
Similar content being viewed by others
References
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)
Dawson D.A., Li Z., Wang H.: Superprocesses with dependent spatial motion and general branching densities. Electronic J. Probab. 6, 1–33 (2001)
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)
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)
Konno N., Shiga T.: Stochasitc partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79, 201–225 (1988)
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)
Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge (1990)
Nualart, D.: The Malliavin calculus and related topics, 2nd edition. Springer (2006)
Reimers M.: One-dimensional stochastic partial differerntial equations and the branching measure diffusion. Probab. Theory Related Fields 81, 319–340 (1989)
Wang H.: State classification for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Related Fields 109, 39–55 (1997)
Wang H.: A class of measure-valued branching diffusions in a random medium. Stoch. Anal. Appl. 16, 753–786 (1998)
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-012-0419-2
Keywords
- Nonlinear stochastic partial differential equation
- Stochastic heat kernel
- Conditional transition probability density in a random environment
- Malliavin calculus
- Hölder continuity
- Moment estimates