Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Intermittency and ageing for the symbiotic branching model

Frank Aurzada and Leif Döring

Full-text: Open access

Abstract

For the symbiotic branching model introduced in [Stochastic Process. Appl. 114 (2004) 127–160], it is shown that ageing and intermittency exhibit different behaviour for negative, zero, and positive correlations. Our approach also provides an alternative, elementary proof and refinements of classical results concerning second moments of the parabolic Anderson model with Brownian potential. Some refinements to more general (also infinite range) kernels of recent ageing results of [Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 461–480] for interacting diffusions are given.

Résumé

Dans le cadre du modèle de branchement symbiotique introduit dans [Stochastic Process. Appl. 114 (2004) 127–160], nous montrons que le vieillissement et l’intermittence présentent différents comportements suivant les cas où la corrélation est négative, positive ou nulle. Notre approche permet de prouver et d’affiner de manière élémentaire des résultats classiques concernant les seconds moments du modèle parabolique d’Anderson avec potentiel Brownien. Nous raffinons aussi quelques résultats récents de vieillissement pour des diffusions interactives à noyaux généraux à portée infinie.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 376-394.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887274

Digital Object Identifier
doi:10.1214/09-AIHP356

Mathematical Reviews number (MathSciNet)
MR2814415

Zentralblatt MATH identifier
1222.60075

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J55: Local time and additive functionals

Keywords
Ageing Interacting diffusions Intermittency Mutually catalytic branching model Parabolic Anderson model Symbiotic branching model

Citation

Aurzada, Frank; Döring, Leif. Intermittency and ageing for the symbiotic branching model. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 376--394. doi:10.1214/09-AIHP356. https://projecteuclid.org/euclid.aihp/1300887274


Export citation

References

  • [1] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge, 1989.
  • [2] J. Blath, L. Döring and A. Etheridge. On the moments and the interface of symbiotic branching model. Preprint, 2009.
  • [3] R. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) viii+125.
  • [4] J. T. Cox, D. A. Dawson and A. Greven. Mutually catalytic super branching random walks: Large finite systems and renormalization analysis. Mem. Amer. Math. Soc. 171 (2004) viii+97.
  • [5] J. T. Cox and A. Klenke. Recurrence and ergodicity of interacting particle systems. Probab. Theory Related Fields 116 (2000) 239–255.
  • [6] D. A. Dawson and E. A. Perkins. Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26 (1998) 1088–1138.
  • [7] A. Dembo and J.-D. Deuschel. Ageing for interacting diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 461–480.
  • [8] A. M. Etheridge and K. Fleischmann. Compact interface property for symbiotic branching. Stochastic Process. Appl. 114 (2004) 127–160.
  • [9] M. Foondun and D. Khoshnevisan. Intermittency for nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 (2009) 548–568.
  • [10] J. Gärtner and S. Molchanov. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 (1990) 613–655.
  • [11] A. Greven and F. den Hollander. Phase transitions for the long-time behavior of interacting diffusions. Ann. Probab. 35 (2007) 1250–1306.
  • [12] B. Hughes. Random Walks and Random Environments, Vol. 1. Oxford Univ. Press, New York, 1995.
  • [13] M. B. Marcus and J. Rosen. Moment generating functions for local times of symmetric Markov processes and random walks. In Probability in Banach Spaces, 8 (Brunswick, ME, 1991) 364–376. Progr. Probab. 30. Birkhäuser, Boston, MA, 1992.
  • [14] T. Shiga. Stepping stone models in population genetics and population dynamics. In Stochastic Processes in Physics and Engineering (Bielefeld, 1986) 345–355. Math. Appl. 42. Reidel, Dordrecht, 1988.
  • [15] T. Shiga and A. Shimizu. Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 (1980) 395–416.