We consider a model of branching Brownian motions in random environment associated with the Poisson random measure. We find a relation between the slow population growth and the localization property in terms of the replica overlap. Applying this result, we prove that, if the randomness of the environment is strong enough, this model possesses the strong localization property, that is, particles gather together at small sets.
References
K. B. Athreya and P. E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972. MR373040K. B. Athreya and P. E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972. MR373040
M. Birkner, J. Geiger and G. Kersting, Branching processes in random environment---a view on critical and subcritical cases, Interacting stochastic systems, 269--291, Springer, Berlin, 2005. MR2118578 1084.60062 10.1007/3-540-27110-4_12M. Birkner, J. Geiger and G. Kersting, Branching processes in random environment---a view on critical and subcritical cases, Interacting stochastic systems, 269--291, Springer, Berlin, 2005. MR2118578 1084.60062 10.1007/3-540-27110-4_12
P. Carmona and Y. Hu, On the partition function of a directed polymer in a Gaussian random environment, Probab. Theory Related Fields 124 (2002), 431--457. MR1939654 1015.60100 10.1007/s004400200213P. Carmona and Y. Hu, On the partition function of a directed polymer in a Gaussian random environment, Probab. Theory Related Fields 124 (2002), 431--457. MR1939654 1015.60100 10.1007/s004400200213
P. Carmona and Y. Hu, Strong disorder implies strong localization for directed polymers in a random environment, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 217--229. MR2249669 1111.60069P. Carmona and Y. Hu, Strong disorder implies strong localization for directed polymers in a random environment, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 217--229. MR2249669 1111.60069
F. Comets, T. Shiga and N. Yoshida, Directed polymers in a random environment: path localization and strong disorder, Bernoulli 9 (2003), 705--723. MR1996276 10.3150/bj/1066223275 euclid.bj/1066223275
F. Comets, T. Shiga and N. Yoshida, Directed polymers in a random environment: path localization and strong disorder, Bernoulli 9 (2003), 705--723. MR1996276 10.3150/bj/1066223275 euclid.bj/1066223275
F. Comets and N. Yoshida, Brownian directed polymers in random environment, Comm. Math. Phys. 254 (2005), 257--287. MR2117626 1128.60089 10.1007/s00220-004-1203-7F. Comets and N. Yoshida, Brownian directed polymers in random environment, Comm. Math. Phys. 254 (2005), 257--287. MR2117626 1128.60089 10.1007/s00220-004-1203-7
S. W. He, J. G. Wang and J. A. Yan, Semimartingale theory and stochastic calculus, Science Press, Beijing, 1992. MR1219534 0781.60002S. W. He, J. G. Wang and J. A. Yan, Semimartingale theory and stochastic calculus, Science Press, Beijing, 1992. MR1219534 0781.60002
Y. Hu and N. Yoshida, Localization for branching random walks in random environment, Stochastic Process. Appl. 119 (2009), 1632--1651. MR2513122 1161.60341 10.1016/j.spa.2008.08.005Y. Hu and N. Yoshida, Localization for branching random walks in random environment, Stochastic Process. Appl. 119 (2009), 1632--1651. MR2513122 1161.60341 10.1016/j.spa.2008.08.005
N. Kaplan, A continuous time Markov branching model with random environments, Advances in Appl. Probability 5 (1973), 37--54. MR339348 0263.60037 10.2307/1425963N. Kaplan, A continuous time Markov branching model with random environments, Advances in Appl. Probability 5 (1973), 37--54. MR339348 0263.60037 10.2307/1425963
Y. Shiozawa, Central limit theorem for branching Brownian motions in random environment, J. Stat. Phys. 136 (2009), 145--163. MR2525233 1171.60024 10.1007/s10955-009-9774-5Y. Shiozawa, Central limit theorem for branching Brownian motions in random environment, J. Stat. Phys. 136 (2009), 145--163. MR2525233 1171.60024 10.1007/s10955-009-9774-5
W. Smith and W. Wilkinson, On branching processes in random environments, Ann. Math. Statist. 40 (1969), 814--827. MR246380 0184.21103 10.1214/aoms/1177697589 euclid.aoms/1177697589
W. Smith and W. Wilkinson, On branching processes in random environments, Ann. Math. Statist. 40 (1969), 814--827. MR246380 0184.21103 10.1214/aoms/1177697589 euclid.aoms/1177697589
N. Yoshida, Central limit theorem for branching random walks in random environment, Ann. Appl. Probab. 18 (2008), 1619--1635. MR2434183 1145.60054 10.1214/07-AAP500 euclid.aoap/1216677134
N. Yoshida, Central limit theorem for branching random walks in random environment, Ann. Appl. Probab. 18 (2008), 1619--1635. MR2434183 1145.60054 10.1214/07-AAP500 euclid.aoap/1216677134