Non-autonomous interacting particle systems in continuum
DOI:
Keywords:
Interacting particle systems, Feller evolution, pure jump process, configuration space, Foster-Lyapunov criterion, Kolmogorov equationAbstract
A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.References
Luisa Arlotti, Bertrand Lods, and Mustapha Mokhtar-Kharroubi, Non-autonomous honesty theory in abstract state spaces with applications to linear kinetic equations, Commun. Pure Appl. Anal. 13 (2014), no. 2, 729-771. MathSciNet CrossRef
Jacek Banasiak and Luisa Arlotti, Perturbations of positive semigroups with applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006. MathSciNet
Jacek Banasiak, Miroslaw Lachowicz, and Marcin Moszynski, Semigroups for generalized birth-and-death equations in $l^ p$ spaces, Semigroup Forum 73 (2006), no. 2, 175-193. MathSciNet CrossRef
V. Bezborodov, Markov birth-and-death dynamics of populations, (2015), arXiv:1502.06783
V. Bezborodov, Spatial birth-and-death Markov dynamics of finite particle systems, (2015), arXiv:1507.05804
B. Bolker, S. Cornell, D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and O. Ovaskainen, A general mathematical framework for the analysis of spatio-temporal point processes, Theoretical Ecology 7 (2014), no. 1, 101-113.
B. Bolker and S. W. Pacala, Using moment equations to understand stochastically driven spatial pattern formation in ecological systems, Theoretical population biology 52 (1997), 179-197.
B. Bolker and S. W. Pacala, Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal, The American Naturalist 153 (1999), no. 6, 575-602.
Mu Fa Chen, Coupling for jump processes, Acta Math. Sinica (N.S.) 2 (1986), no. 2, 123-136. MathSciNet CrossRef
Mu-Fa Chen, From Markov chains to non-equilibrium particle systems, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MathSciNet CrossRef
C. Costantini, A. Gerardi, and G. Nappo, On the convergence of sequences of stationary jump Markov processes, Statist. Probab. Lett. 1 (1983), no. 3, 155-160. MathSciNet CrossRef
C. Costantini and G. Nappo, Some results on weak convergence of jump Markov processes and their stability properties, Systems Control Lett. 2 (1982/83), no. 3, 175-183. MathSciNet CrossRef
U. Dieckmann and R. Law, Moment Approximations of Individual-based Models, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge University Press, 2000, pp. 252-270.
U. Dieckmann and R. Law, Relaxation projections and the method of moments, The Geometry of Ecological Interactions: Symplifying Spatial Complexity, Cambridge University Press, 2005, pp. 412-455.
Andreas Eibeck and Wolfgang Wagner, Stochastic particle approximations for Smoluchoskis coagulation equation, Ann. Appl. Probab. 11 (2001), no. 4, 1137-1165. MathSciNet CrossRef
Andreas Eibeck and Wolfgang Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab. 13 (2003), no. 3, 845-889. MathSciNet CrossRef
Eugene A. Feinberg, Manasa Mandava, and Albert N. Shiryaev, On solutions of Kolmogorovs equations for nonhomogeneous jump Markov processes, J. Math. Anal. Appl. 411 (2014), no. 1, 261-270. MathSciNet CrossRef
William Feller, An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MathSciNet
William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MathSciNet
Willy Feller, On the integro-differential equations of purely discontinuous Markoff processes, Trans. Amer. Math. Soc. 48 (1940), 488-515. MathSciNet
D. Finkelshtein, M. Friesen, H. Hatzikirou, Y. Kondratiev, T. Kruger, and O. Kutoviy, Stochastic models of tumour development and related mesoscopic equations, Inter. Stud. Comp. Sys. 7 (2015), 5-85.
Nicolas Fournier and Sylvie Meleard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab. 14 (2004), no. 4, 1880-1919. MathSciNet CrossRef
I. I. Gihman and A. V. Skorohod, The theory of stochastic processes. II, Springer-Verlag, New York-Heidelberg, 1975. MathSciNet
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society, Providence, R. I., 1974. MathSciNet
Tosio Kato, On the semi-groups generated by Kolmogoroffs differential equations, J. Math. Soc. Japan 6 (1954), 1-15. MathSciNet
M. Kimura and T. Maruyama, The mutational load with epistatic gene interaction, Genetics 54 (1966), 1337-1351.
V. N. Kolokoltsov, Kinetic equations for the pure jump models of $k$-nary interacting particle systems, Markov Process. Related Fields 12 (2006), no. 1, 95-138. MathSciNet
Thomas M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005. MathSciNet
Sean P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab. 25 (1993), no. 3, 518-548. MathSciNet CrossRef
Claudia Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc. 48 (2001), no. 11, 1304-1314. MathSciNet
David Steinsaltz, Steven N. Evans, and Kenneth W. Wachter, A generalized model of mutation-selection balance with applications to aging, Adv. in Appl. Math. 35 (2005), no. 1, 16-33. MathSciNet CrossRef
H. R. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, Positivity IV---theory and applications, Tech. Univ. Dresden, Dresden, 2006, pp. 135-146. MathSciNet
J. A. van Casteren, Markov processes, Feller semigroups and evolution equations, Series on Concrete and Applicable Mathematics, vol. 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MathSciNet
Shao Yi Zhang, Necessary and sufficient conditions for the existence of a successful coupling for jump processes, Acta Math. Appl. Sinica 22 (1999), no. 2, 231-235. MathSciNet
J. Zheng and X. Zheng, A martingale approach to Q-processes, Kexue Tongbao 32 (1987), no. 21, 1457-1459.
