×

Some properties of superprocesses under a stochastic flow. (English) Zbl 1171.60011

Summary: For a superprocess under a stochastic flow in one dimension, we prove that it has a density with respect to the Lebesgue measure. A stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov’s \(L_p\)-theory for linear SPDE.

MSC:

60G57 Random measures
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML

References:

[1] D. A. Dawson, Z. Li and H. Wang. Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6 (2001) 1-33. · Zbl 1008.60093
[2] D. A. Dawson, J. Vaillancourt and H. Wang. Stochastic partial differential equations for a class of interacting measure-valued diffusions. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 167-180. · Zbl 0973.60077
[3] A. Friedman. Stochastic Differential Equations and Applications , Vol. 1. Academic Press, New York, 1975. · Zbl 0323.60056
[4] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes . North Holland/Kodansha, Amsterdam, 1989. · Zbl 0684.60040
[5] G. Kallianpur and J. Xiong. Stochastic differential equations on infinite dimensional spaces. IMS Lecture Notes-Monograph Series , Vol. 26, 1995. · Zbl 0845.60008
[6] N. Konno and T. Shiga. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 (1988) 201-225. · Zbl 0631.60058
[7] N. V. Krylov. An analytic approach to SPDEs, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr. 64 185-242. Amer. Math. Soc., Providence, RI, 1999. · Zbl 0933.60073
[8] G. Skoulakis and R. J. Adler. Superprocesses over a stochastic flow. Ann. Appl. Probab. 11 (2001) 488-543. · Zbl 1018.60052
[9] J. B. Walsh. An Introduction to Stochastic Partial Differential Equations . In École d’été de probabilités de Saint-Flour , XIV-1984 256-439. Lecture Notes in Math. 1180 . Springer-Verlag, Berlin, 1986. · Zbl 0608.60060
[10] H. Wang. State classification for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Related Fields 109 (1997) 39-55. · Zbl 0882.60092
[11] H. Wang. A class of measure-valued branching diffusions in a random medium. Stochastic Anal. Appl. 16 (1998) 753-786. · Zbl 0913.60091
[12] J. Xiong. A stochastic log-Laplace equation. Ann. Probab. 32 (2004) 2362-2388. · Zbl 1055.60042
[13] J. Xiong. Long-term behavior for superprocesses over a stochastic flow. Electron. Comm. Probab. 9 (2004) 36-52. · Zbl 1060.60084
[14] J. Xiong and X. Zhou. Superprocess over a stochastic flow with superprocess catalyst. Internat. J. Pure Appl. Mathematics 17 (2004) 353-382. · Zbl 1063.60123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.