Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Branching random walks in random environment and super-Brownian motion in random environment

Makoto Nakashima

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Abstract

We focus on the existence and characterization of the limit for a certain critical branching random walks in time–space random environment in one dimension which was introduced by Birkner, Geiger and Kersting in (In Interacting Stochastic Systems (2005) 269–291 Springer). Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time–space site. The limit of this measure-valued processes is characterized as the unique solution to the non-trivial martingale problem and called super-Brownian motion in a random environment by Mytnik in (Ann. Probab. 24 (1996) 1953–1978).

Résumé

Nous étudions l’existence et la caractérisation de la limite de marches branchantes critiques dans un environnement spatio-temporel aléatoire en dimension 1 introduit par Birkner, Geiger and Kersting dans (In Interacting Stochastic Systems (2005) 269–291 Springer). Chaque particule effectue une marche aléatoire simple sur $\mathbb{Z}$ et le mécanisme de branchement dépend du site indexé par l’espace et le temps. La limite de ce processus à valeur mesure est caractérisée comme l’unique solution d’un problème de martingale non-trivial et correspond au super mouvement Brownien en environnement aléatoire par Mytnik dans (Ann. Probab. 24 (1996) 1953–1978).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1251-1289.

Dates
Received: 30 June 2013
Revised: 16 April 2014
Accepted: 16 April 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432042

Digital Object Identifier
doi:10.1214/14-AIHP620

Mathematical Reviews number (MathSciNet)
MR3414447

Zentralblatt MATH identifier
1329.60358

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60J68: Superprocesses 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments

Keywords
Superprocesses in a random environment Branching random walks in a random environment Stochastic heat equations Uniqueness

Citation

Nakashima, Makoto. Branching random walks in random environment and super-Brownian motion in random environment. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1251--1289. doi:10.1214/14-AIHP620. https://projecteuclid.org/euclid.aihp/1445432042


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