Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-08-22T03:37:06.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems of occupation times of normalized binary contact path processes on lattices

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  27 August 2024

Xiaofeng Xue
Xiaofeng Xue*
Affiliation:
Beijing Jiaotong University
*
*Postal address: School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China. Email: xfxue@bjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with limit theorems of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We first show that the law of large numbers of the occupation time process is driven by the identity function when the dimension of the lattice is at least 3 and the infection rate of the model is sufficiently large conditioned on the initial state of the NBCPP being distributed with a particular invariant distribution. Then we show that the centered occupation time process of the NBCPP converges in finite-dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with the aforementioned invariant distribution.

Keywords

Binary contact path processoccupation timecentral limit theorem

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 44 - 66
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Birkner, M. and Zähle, I. (2007). A functional CLT for the occupation time of a state-dependent branching random walk. Ann. Prob. 35, 2063–2090.CrossRefGoogle Scholar
Bramson, M., Cox, J. T. and Griffeath, D. (1988). Occupation time large deviations of the voter model. Prob. Theory Relat. Fields 77, 401–413.CrossRefGoogle Scholar
Cox, J. T. and Griffeath, D. (1983). Occupation time limit theorems for the voter model. Ann. Prob. 11, 876–893.CrossRefGoogle Scholar
Deuschel, J. D. and Rosen, J. (1998). Occupation time large deviations for critical branching Brownian motion and related processes. Ann. Prob. 26, 602–643.CrossRefGoogle Scholar
Ethier, N. and Kurtz, T. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Griffeath, D. (1983). The binary contact path process. Ann. Prob. 11, 692–705.CrossRefGoogle Scholar
Komorowski, T., Landim, C. and Olla, S. (2012). Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Springer, New York.CrossRefGoogle Scholar
Landim, C. (1992). Occupation time large deviations for the symmetric simple exclusion process. Ann. Prob. 20, 206–231.CrossRefGoogle Scholar
Lee, T. Y., Landim, C. and Chang, C. (2004). Occupation time large deviations of two-dimensional symmetric simple exclusion process. Ann. Prob. 32, 661–691.Google Scholar
Li, Q. and Ren, Y. (2011). A large deviation for occupation time of critical branching $\alpha$ -stable process. Sci. China Math. 54, 1445–1456.CrossRefGoogle Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York.CrossRefGoogle Scholar
Maillard, G. and Mountford, T. (2009). Large deviations for voter model occupation times in two dimensions. Ann. Inst. H. Poincaré Prob. Statist. 45, 577–588.CrossRefGoogle Scholar
Nakashima, M. (2009). Central limit theorem for linear stochastic evolutions. J. Math. Kyoto U. 49, 201–224.Google Scholar
Newman, C. (1983). A general central limit theorem for FKG systems. Commun. Math. Phys. 91, 75–80.CrossRefGoogle Scholar
Schonmann, H. (1986). Central limit theorem for the contact process. Ann. Prob. 14, 1291–1295.CrossRefGoogle Scholar
Xue, X. (2018). An improved upper bound for critical value of the contact process on $\mathbb{Z}^d$ with $d\geq 3$ . Electron. Commun. Prob. 23, 1–11.CrossRefGoogle Scholar
Xue, X. and Zhao, L. (2021). Non-equilibrium fluctuations of the weakly asymmetric normalized binary contact path process. Stoch. Process. Appl. 135, 227–253.CrossRefGoogle Scholar