Nakashima, Makoto
Branching random walks in random environment and super-brownian motion in random environment
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4 , p. 1251-1289
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
MR 3414447
doi : 10.1214/14-AIHP620
URL stable : http://www.numdam.org/item?id=AIHPB_2015__51_4_1251_0

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 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).
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 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).

Bibliographie

[1] L. Bertini and G. Giacomin. Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 (3) (1997) 571–607. MR 1462228 | Zbl 0874.60059

[2] M. Birkner, J. Geiger and G. Kersting. Branching processes in random environment: A view on critical and subcritical cases. In Interacting Stochastic Systems 269–291. Springer, Berlin, 2005. MR 2118578 | Zbl 1084.60062

[3] D. L. Burkholder. Distribution function inequalities for martingales. Ann. Probab. 1 (1973) 19–42. MR 365692 | Zbl 0301.60035

[4] D. A. Dawson. Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5 (1) (1975) 1–52. MR 388539 | Zbl 0299.60050

[5] D. A. Dawson. Geostochastic calculus. Canad. J. Statist. 6 (2) (1978) 143–168. MR 532855 | Zbl 0399.60047

[6] D. A. Dawson. Measure-valued Markov processes. In École d’Été de Probabilités de Saint-Flour XXI – 1991 1–260. Lecture Notes in Math. 1541. Springer, Berlin, 1993. MR 1242575 | Zbl 0799.60080

[7] D. A. Dawson and E. A. Perkins. Historical Processes. Mem. Amer. Math. Soc. 93 No. 454. Amer. Math. Soc., Providence, RI, 1991. MR 1079034 | Zbl 0754.60062

[8] E. B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society Colloquium Publications 50. Amer. Math. Soc., Providence, RI, 2002. MR 1883198 | Zbl 0999.60003

[9] E. B. Dynkin. Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series 34. Amer. Math. Soc., Providence, RI, 2004. Appendix A by J.-F. Le Gall and Appendix B by I. E. Verbitsky. MR 2089791 | Zbl 1079.60006

[10] A. M. Etheridge. An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI, 2000. MR 1779100 | Zbl 0971.60053

[11] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, Hoboken, NJ, 2009. MR 838085 | Zbl 1089.60005

[12] H. Heil and M. Nakashima. A remark on localization for branching random walks in random environment. Electron. Commun. Probab. 16 (2011) 323–336. MR 2819656 | Zbl 1225.60158

[13] H. Heil, M. Nakashima and N. Yoshida. Branching random walks in random environment are diffusive in the regular growth phase. Electron. J. Probab. 16 (2011) 1318–1340. MR 2827461 | Zbl 1244.60100

[14] N. Konno and T. Shiga. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 (2) (1988) 201–225. MR 958288 | Zbl 0631.60058

[15] J. F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1999. MR 1714707 | Zbl 0938.60003

[16] J. F. Le Gall, E. A. Perkins and S. J. Taylor. The packing measure of the support of super-Brownian motion. Stochastic Process. Appl. 59 (1) (1995) 1–20. MR 1350253 | Zbl 0848.60078

[17] C. Mueller, L. Mytnik and E. A. Perkins. Nonuniqueness for a parabolic SPDE with 3/4-ε-Hölder diffusion coefficients. Ann. Probab. 42 (2014) 2032–2112. MR 3262498 | Zbl 1301.60080

[18] C. Mueller and E. A. Perkins. The compact support property for solutions to the heat equation with noise. Probab. Theory Related Fields 44 (1992) 325–358. MR 1180704 | Zbl 0767.60054

[19] L. Mytnik. Superprocesses in random environments. Ann. Probab. 24 (4) (1996) 1953–1978. MR 1415235 | Zbl 0874.60041

[20] L. Mytnik. Weak uniqueness for the heat equation with noise. Ann. Probab. 26 (3) (1998) 968–984. MR 1634410 | Zbl 0935.60045

[21] L. Mytnik and E. A. Perkins. Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: The white noise case. Probab. Theory Related Fields 149 (1–2) (2011) 1–96. MR 2773025 | Zbl 1233.60039

[22] M. Nakashima. Almost sure central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 21 (1) (2011) 351–373. MR 2759206 | Zbl 1210.60108

[23] E. A. Perkins. A space–time property of a class of measure-valued branching diffusions. Trans. Amer. Math. Soc. 305 (2) (1988) 743–795. MR 924777 | Zbl 0641.60060

[24] E. A. Perkins. The Hausdorff measure of the closed support of super-Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 25 (2) (1989) 205–224. Numdam | MR 1001027 | Zbl 0679.60053

[25] E. A. Perkins. Part II: Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics 125–329. Springer, Berlin, 2002. MR 1915445 | Zbl 1020.60075

[26] M. Reimers. One dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 (3) (1989) 319–340. MR 983088 | Zbl 0651.60069

[27] T. Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46 (2) (1994) 415–437. MR 1271224 | Zbl 0801.60050

[28] Y. Shiozawa. Central limit theorem for branching Brownian motions in random environment. J. Stat. Phys. 136 (1) (2009) 145–163. MR 2525233 | Zbl 1171.60024

[29] Y. Shiozawa. Localization for branching Brownian motions in random environment. Tohoku Math. J. (2) 61 (4) (2009) 483–497. MR 2598246 | Zbl 1187.60092

[30] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. Kyoto J. Math. 8 (1) (1968) 141–167. MR 237008 | Zbl 0159.46201

[31] N. Yoshida. Central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 18 (4) (2008) 1619–1635. MR 2434183 | Zbl 1145.60054