Superprocesses in random environments. (English) Zbl 0874.60041

The paper establishes the weak convergence of a sequence of branching particle systems whose branching mechanism is affected by a random environment. The environment is specified by a sequence \(\xi_k\), \(k=1,2,\dots\), of independent, identically distributed zero-mean random fields on the space \(E\) in which the particles live. The \(n\)th particle system \(X^n_t\), \(t\geq 0\), starts at time \(t=0\) with \(K_n\) particles \((K_n\sim n)\) and during any time interval \((k/n,(k+1)/n)\), \(k=0,1,2,\dots\), all existing particles move independently of each other in accordance with a Feller process on \(E\). Conditionally on \(\xi^{(n)}_k\) (where \(k\geq 1\) and \(\xi^{(n)}_k\) is \(\xi_k\) truncated at \(\pm\sqrt n\)), each particle existing at time \(k/n\) either splits into two particles or dies, with probabilities \({1\over 2}+{1\over 2\sqrt n} \xi^{(n)}_k(x)\) and \({1\over 2}-{1\over 2\sqrt n} \xi^{(n)}_k(x)\), respectively, independently of other particles. Here \(x\) denotes the location of the particle. The author proves that if \(X^n_0\) converges, then the sequence \(X^n\), \(n=1,2,\dots\), converges weakly to a measure-valued superprocess which is the unique solution of a certain martingale problem.


60G57 Random measures
60F17 Functional limit theorems; invariance principles
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI


[1] CARMONA, R. and MOLCHANOV, S. 1994. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 518. · Zbl 0925.35074
[2] DAWSON, D. 1980. Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141 180. \' \' · Zbl 0439.60051 · doi:10.1016/0047-259X(80)90012-3
[3] DAWSON, D. 1993. Measure-valued processes. Ecole d’Ete de Probabilities de Saint Flour. \' Ĺecture Notes in Math. 1541 1 26. Springer, New York. · Zbl 0799.60080
[4] DAWSON, D. and FLEISCHMANN, K. 1990. Critical branching in a highly fluctuating random medium. Probab. Theory Related Fields 90 241 274. · Zbl 0735.60087 · doi:10.1007/BF01192164
[5] DAWSON, D. and PERKINS, E. 1991. Historical processes. Mem. Amer. Math. Soc. 454. · Zbl 0754.60062
[6] Dy NKIN, E. 1993. Superprocesses and partial differential equations. Ann. Probab. 21 1184 1262. · Zbl 0806.60066 · doi:10.1214/aop/1176989116
[7] ETHIER, S. N. and KURTZ, T. G. 1986. Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[8] FLEISCHMANN, K. and MOLCHANOV, S. 1990. Exact asy mptotics in a mean field model with random potential. Probab. Theory Related Fields 86 239 251. · Zbl 0677.60105 · doi:10.1007/BF01474644
[9] GARTNER, J. and MOLCHANOV, S. 1990. Parabolic problems for the Anderson model. Comm. Math. Phy s. 132 613 655. · Zbl 0711.60055 · doi:10.1007/BF02156540
[10] HELLAND, I. 1981. Minimal conditions for weak convergence to a diffusion process on the line. Ann. Probab. 9 429 452. · Zbl 0459.60027 · doi:10.1214/aop/1176994416
[11] IKEDA, N. and WATANABE, S. 1989. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. · Zbl 0684.60040
[12] JACOD, J. and SHIRy AEV, A. 1987. Limit Theorems for Stochastic Processes. Springer, New York.
[13] KEIDING, N. 1975. Extinction and exponential growth in random environments. Theoret. Population Biol. 8 49 63. · Zbl 0311.92019 · doi:10.1016/0040-5809(75)90038-6
[14] KURTZ, T. G. 1981. Approximation of Population Processes. SIAM, Philadelphia. · Zbl 0465.60078
[15] MELEARD, S. and ROELLY, S. 1993. Interacting measure branching processes. Some bounds for the support. Stochastics and Stochastics Reports 44 103 121. · Zbl 0786.60065
[16] MUELLER, C. and PERKINS, E. 1992. The compact support property for solutions to the heat equation with noise. Probab. Theory Related Fields 93 325 358. · Zbl 0767.60054 · doi:10.1007/BF01193055
[17] PERKINS, E. 1992. Measure-valued branching diffusions with spatial interactions. Probab. Theory Related Fields 94 189 245. · Zbl 0767.60044 · doi:10.1007/BF01192444
[18] PERKINS, E. 1995. On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 549. · Zbl 0823.60071
[19] SAMORODNITSKY, G. and TAQQU, M. 1994. Stable Non-Gaussian Random Processes. Chapman and Hall, New York. · Zbl 0925.60027
[20] WALSH, J. 1986. An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 265 439. Springer, New York. · Zbl 0608.60060
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.