The Annals of Probability

A stochastic log-Laplace equation

Jie Xiong

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Abstract

We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103–126]. We also derive the Wong–Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488–543].

Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2362-2388.

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813616

Digital Object Identifier
doi:10.1214/009117904000000540

Mathematical Reviews number (MathSciNet)
MR2078543

Zentralblatt MATH identifier
1055.60042

Subjects
Primary: 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Superprocess random environment Wong–Zakai approximation particle system representation stochastic partial differential equation

Citation

Xiong, Jie. A stochastic log-Laplace equation. Ann. Probab. 32 (2004), no. 3B, 2362--2388. doi:10.1214/009117904000000540. https://projecteuclid.org/euclid.aop/1091813616


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