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.
Electron. J. Probab.
19:
1-46
(2014).
DOI: 10.1214/EJP.v19-2904
Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489–526. MR874906 10.1007/BF01212322 euclid.cmp/1104116538
Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489–526. MR874906 10.1007/BF01212322 euclid.cmp/1104116538
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. MR2372887Aizenman, 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. MR762034 10.1007/BF01015729Aizenman, Michael; Newman, Charles M. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 (1984), no. 1-2, 107–143. MR762034 10.1007/BF01015729
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 MR1118840Anderson, 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 10.1007/s00440-009-0214-xAthreya, 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 10.1007/s00440-009-0214-x
Bezuidenhout, Carol; Grimmett, Geoffrey. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991), no. 3, 984–1009. MR1112404 10.1214/aop/1176990332 euclid.aop/1176990332
Bezuidenhout, Carol; Grimmett, Geoffrey. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991), no. 3, 984–1009. MR1112404 10.1214/aop/1176990332 euclid.aop/1176990332
Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1999), no. 1, 29–66. MR1675890 10.1007/s000390050080Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1999), no. 1, 29–66. MR1675890 10.1007/s000390050080
Darroch, J. N.; Seneta, E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probability 4 1967 192–196. MR212866 10.2307/3212311Darroch, J. N.; Seneta, E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probability 4 1967 192–196. MR212866 10.2307/3212311
Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999–1040. MR757768 10.1214/aop/1176993140 euclid.aop/1176993140
Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999–1040. MR757768 10.1214/aop/1176993140 euclid.aop/1176993140
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 10.1214/aop/1023481003 euclid.aop/1023481003
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 10.1214/aop/1023481003 euclid.aop/1023481003
Ferrari, P. A.; Kesten, H.; MartÃnez, 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 10.1214/aoap/1034968146 euclid.aoap/1034968146
Ferrari, P. A.; Kesten, H.; MartÃnez, 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 10.1214/aoap/1034968146 euclid.aoap/1034968146
Fleischmann, Klaus; Swart, Jan M. Trimmed trees and embedded particle systems. Ann. Probab. 32 (2004), no. 3A, 2179–2221. MR2073189 10.1214/009117904000000090 euclid.aop/1089808423
Fleischmann, Klaus; Swart, Jan M. Trimmed trees and embedded particle systems. Ann. Probab. 32 (2004), no. 3A, 2179–2221. MR2073189 10.1214/009117904000000090 euclid.aop/1089808423
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 MR1707339Grimmett, 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 10.1214/EJP.v4-49Greven, 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 10.1214/EJP.v4-49
Häggström, Olle. Percolation beyond $\Bbb Z^ d$: the contributions of Oded Schramm. Ann. Probab. 39 (2011), no. 5, 1668–1701. MR2884871 10.1214/10-AOP563 euclid.aop/1318940779
Häggström, Olle. Percolation beyond $\Bbb Z^ d$: the contributions of Oded Schramm. Ann. Probab. 39 (2011), no. 5, 1668–1701. MR2884871 10.1214/10-AOP563 euclid.aop/1318940779
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 10.1214/EJP.v15-783 euclid.ejp/1464819812
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 10.1214/EJP.v15-783 euclid.ejp/1464819812
Kallenberg, Olav. Stability of critical cluster fields. Math. Nachr. 77 (1977), 7–43. MR443078 10.1002/mana.19770770102Kallenberg, Olav. Stability of critical cluster fields. Math. Nachr. 77 (1977), 7–43. MR443078 10.1002/mana.19770770102
Kesten, Harry. Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25 (1981), no. 4, 717–756. MR633715 10.1007/BF01022364Kesten, Harry. Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25 (1981), no. 4, 717–756. MR633715 10.1007/BF01022364
Kingman, J. F. C. The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13 1963 337–358. MR152014 10.1112/plms/s3-13.1.337Kingman, J. F. C. The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13 1963 337–358. MR152014 10.1112/plms/s3-13.1.337
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 MR974565Liemant, 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 MR974565
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 MR776231Liggett, 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 MR776231
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 MR1717346Liggett, 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 MR2574430Liggett, 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
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 MR1600720Norris, 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 10.2307/1427497Nair, 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 10.2307/1427497
Swart, Jan M. The contact process seen from a typical infected site. J. Theoret. Probab. 22 (2009), no. 3, 711–740. MR2530110 10.1007/s10959-008-0184-4Swart, Jan M. The contact process seen from a typical infected site. J. Theoret. Probab. 22 (2009), no. 3, 711–740. MR2530110 10.1007/s10959-008-0184-4
Vere-Jones, D. Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22 1967 361–386. MR214145 10.2140/pjm.1967.22.361 euclid.pjm/1102992207
Vere-Jones, D. Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22 1967 361–386. MR214145 10.2140/pjm.1967.22.361 euclid.pjm/1102992207