Subcritical contact processes seen from a typical infected site

Anja Sturm (University of Göttingen)
Jan M. Swart (Institute of Information Theory and Automation of the ASCR)

Abstract


What is the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice? This question is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a time-dependent exponential factor. In this paper, we study eigenmeasures of contact processes on general countable groups in the subcritical regime. We prove that in this regime, the process has a unique spatially homogeneous eigenmeasure. As an application, we show that the law of the process as seen from a typical infected site, chosen according to a Campbell law, converges to a long-time limit. We also show that the exponential decay rate of the expected number of infected sites is continuously differentiable and strictly increasing as a function of the recovery rate, and we give a formula for the derivative in terms of the long time limit law of the process as seen from a typical infected site.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-46

Publication Date: June 23, 2014

DOI: 10.1214/EJP.v19-2904

References

  • Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489--526. MR0874906
  • Aizenman, Michael; Jung, Paul. On the critical behavior at the lower phase transition of the contact process. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 301--320. MR2372887
  • Aizenman, Michael; Newman, Charles M. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 (1984), no. 1-2, 107--143. MR0762034
  • Anderson, William J. Continuous-time Markov chains. An applications-oriented approach. Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York, 1991. xii+355 pp. ISBN: 0-387-97369-9 MR1118840
  • Athreya, Siva R.; Swart, Jan M. Survival of contact processes on the hierarchical group. Probab. Theory Related Fields 147 (2010), no. 3-4, 529--563. MR2639714
  • Bezuidenhout, Carol; Grimmett, Geoffrey. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991), no. 3, 984--1009. MR1112404
  • Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1999), no. 1, 29--66. MR1675890
  • Darroch, J. N.; Seneta, E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probability 4 1967 192--196. MR0212866
  • Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999--1040. MR0757768
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2002), no. 2, 579--604. MR1905852
  • Ferrari, P. A.; Kesten, H.; Martínez, S. $R$-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata. Ann. Appl. Probab. 6 (1996), no. 2, 577--616. MR1398060
  • Fleischmann, Klaus; Swart, Jan M. Trimmed trees and embedded particle systems. Ann. Probab. 32 (2004), no. 3A, 2179--2221. MR2073189
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339
  • Greven, A.; Klenke, A.; Wakolbinger, A. The longtime behavior of branching random walk in a catalytic medium. Electron. J. Probab. 4 (1999), no. 12, 80 pp. (electronic). MR1690316
  • Häggström, Olle. Percolation beyond $\Bbb Z^ d$: the contributions of Oded Schramm. Ann. Probab. 39 (2011), no. 5, 1668--1701. MR2884871
  • van der Hofstad, Remco; Sakai, Akira. Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: the higher-point functions. Electron. J. Probab. 15 (2010), 801--894. MR2653947
  • Kallenberg, Olav. Stability of critical cluster fields. Math. Nachr. 77 (1977), 7--43. MR0443078
  • Kesten, Harry. Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25 (1981), no. 4, 717--756. MR0633715
  • Kingman, J. F. C. The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13 1963 337--358. MR0152014
  • Liemant, A.; Matthes, K.; Wakolbinger, A. Equilibrium distributions of branching processes. Mathematics and its Applications (East European Series), 34. Kluwer Academic Publishers Group, Dordrecht, 1988. 240 pp. ISBN: 90-277-2774-0 MR0974565
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231
  • Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. xii+332 pp. ISBN: 3-540-65995-1 MR1717346
  • Liggett, Thomas M. Continuous time Markov processes. An introduction. Graduate Studies in Mathematics, 113. American Mathematical Society, Providence, RI, 2010. xii+271 pp. ISBN: 978-0-8218-4949-1 MR2574430
  • Menʹshikov, M. V. Coincidence of critical points in percolation problems. (Russian) Dokl. Akad. Nauk SSSR 288 (1986), no. 6, 1308--1311. MR0852458
  • Norris, J. R. Markov chains. Reprint of 1997 original. Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. xvi+237 pp. ISBN: 0-521-48181-3 MR1600720
  • Nair, M. G.; Pollett, P. K. On the relationship between $\mu$-invariant measures and quasi-stationary distributions for continuous-time Markov chains. Adv. in Appl. Probab. 25 (1993), no. 1, 82--102. MR1206534
  • J.M. Swart. Extinction versus unbounded growth. Habilitation Thesis of the University Erlangen-Nürnberg, 2007. arXiv:math/0702095v1.
  • Swart, Jan M. The contact process seen from a typical infected site. J. Theoret. Probab. 22 (2009), no. 3, 711--740. MR2530110
  • Vere-Jones, D. Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22 1967 361--386. MR0214145
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.