Skip to main content Accessibility help
×
Home

Boundary effect in competition processes

  • Vadim Shcherbakov (a1) and Stanislav Volkov (a2)

Abstract

This paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

Copyright

Corresponding author

*Postal address: Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, UK.
***Postal address: Centre for Mathematical Sciences, Lund University, SE-221 00 Lund, Sweden.
****Email address: s.volkov@maths.lth.se

References

Hide All
[1] Anderson, W. (1991). Continuous-Time Markov Chains: An Application-Oriented Approach. Springer, New York.
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
[3] Freedman, D. A. (1965). Bernard Friedman’s urn. Ann. Math. Statist. 36, 956970.
[4] Iglehart, D. L. (1964). Reversible competition processes. Z. Wahrseheinliehkeitsth. 2, 314331.
[5] Iglehart, D. L. (1964). Multivariate competition processes. Ann. Math. Statist. 35, 350361.
[6] Janson, S., Shcherbakov, V. and Volkov, S. (2019). Long-term behaviour of a reversible system of interacting random walks. J. Stat. Phys. 175, 7196.
[7] Kingman, J. F. C. and Volkov, S. E. (2003). Solution to the OK Corral model via decoupling of Friedman’s urn. J. Theoret. Prob. 16, 267276.
[8] MacPhee, M. I., Menshikov, M. V. and Wade, A. R. (2010). Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift. Markov Process. Relat. Fields 16, 351388.
[9] Menshikov, M. V., Popov, S. and Wade, A. R. (2017). Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge University Press.
[10] Menshikov, M. and Shcherbakov, V. (2018). Long-term behaviour of two interacting birth-and-death processes. Markov Process. Relat. Fields. 24, 85106.
[11] Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surv. 4, 179.
[12] Reuter, G. E. H. (1961). Competition processes. In Proc. 4th Berkeley Symp. Math. Statist. Probab., vol. II. University of California Press, Berkeley.
[13] Shcherbakov, V. and Volkov, S. (2015). Long-term behaviour of locally interacting birth-and-death processes. J. Stat. Phys. 158, 132157.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed