A stochastic log-Laplace equation
Jie Xiong
Source: Ann. Probab. Volume 32, Number 3B
(2004), 2362-2388.
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].
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Keywords: Superprocess; random environment; Wong–Zakai approximation; particle system representation; stochastic partial differential equation
Full-text: Open access
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1091813616
Digital Object Identifier: doi:10.1214/009117904000000540
Mathematical Reviews number (MathSciNet): MR2078543
Zentralblatt MATH identifier: 02121699
References
Da Prato, G. and Tubaro, L., eds. (2002). Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. 227. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR899292
Zentralblatt MATH: 0602.00007
Dawson, D. A. (1993). Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI. Lecture Notes in Math. 1541 1--260. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1242575
Zentralblatt MATH: 0799.60080
Dawson, D. A. and Fleischmann, K. (1997). A continuous super-Brownian motion in a super-Brownian medium. J. Theoret. Probab. 10 213--276.
Mathematical Reviews (MathSciNet): MR1432624
Digital Object Identifier: doi:10.1023/A:1022606801625
Zentralblatt MATH: 0877.60030
Dawson, D. A., Li, Z. H. and Wang, H. (2001). Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6 1--33.
Mathematical Reviews (MathSciNet): MR1873302
Dynkin, E. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185--1262.
Mathematical Reviews (MathSciNet): MR1235414
JSTOR: links.jstor.org
Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. Wiley, New York.
Mathematical Reviews (MathSciNet): MR838085
Zentralblatt MATH: 0592.60049
Fleischmann, K. and Xiong, J. (2001). A cyclically catalytic super-Brownian motion. Ann. Probab. 2 820--861.
Mathematical Reviews (MathSciNet): MR1849179
Digital Object Identifier: doi:10.1214/aop/1008956694
Project Euclid: euclid.aop/1008956694
Zentralblatt MATH: 1017.60099
Friedman, A. (1975). Stochastic Differential Equations and Applications 1. Academic Press, New York.
Zentralblatt MATH: 0323.60056
Mathematical Reviews (MathSciNet): MR494490
Gyöngy, I. (2002). Approximations of stochastic partial differential equations. Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. 227 287--307. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1919514
Ibragimov, I. A. (1983). On smoothness conditions for trajectories of random functions. Theory Probab. Appl. 28 240--262.
Mathematical Reviews (MathSciNet): MR700208
Kallianpur, G. and Xiong, J. (1995). Stochastic Differential Equations in Infinite Dimensional Spaces. IMS, Hayward, CA.
Mathematical Reviews (MathSciNet): MR1465436
Kotelenez, P. (1992). Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics Stochastics Rep. 41 177--199.
Mathematical Reviews (MathSciNet): MR1275582
Kotelenez, P. (1992). Comparison methods for a class of function valued stochastic partial differential equations. Probab. Theory Related Fields 93 1--19.
Mathematical Reviews (MathSciNet): MR1172936
Digital Object Identifier: doi:10.1007/BF01195385
Zentralblatt MATH: 0767.60053
Krylov, N. V. (2002). Some new results in the theory of SPDEs in Sobolev spaces. Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. 227 325--336. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1919516
Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 103--126.
Mathematical Reviews (MathSciNet): MR1705602
Digital Object Identifier: doi:10.1016/S0304-4149(99)00024-1
Zentralblatt MATH: 0996.60071
Li, Z. H., Wang, H. and Xiong, J. (2004). Conditional log-Laplace functionals of immigration superprocesses with dependent spatial motion. WIAS Preprint No. 900.
Mathematical Reviews (MathSciNet): MR2169037
Digital Object Identifier: doi:10.1007/s10440-005-6696-3
Rozovskii, B. L. (1990). Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht.
Mathematical Reviews (MathSciNet): MR1135324
Zentralblatt MATH: 0724.60070
Skoulakis, G. and Adler, R. J. (2001). Superprocesses over a stochastic flow. Ann. Appl. Probab. 11 488--543.
Mathematical Reviews (MathSciNet): MR1843056
Digital Object Identifier: doi:10.1214/aoap/1015345302
Project Euclid: euclid.aoap/1015345302
Zentralblatt MATH: 1018.60052
Wang, H. (1998). A class of measure-valued branching diffusions in a random medium. Stochastic Anal. Appl. 16 753--786.
Mathematical Reviews (MathSciNet): MR1632574
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