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, pp. 1251-1289.

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).

DOI : 10.1214/14-AIHP620
Mots clés : superprocesses in a random environment, branching random walks in a random environment, stochastic heat equations, uniqueness
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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, pp. 1251-1289. doi : 10.1214/14-AIHP620. http://archive.numdam.org/articles/10.1214/14-AIHP620/

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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. | DOI | MR | Zbl

[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 | Zbl

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

[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 | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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