Summary
The basic convergence theorems for finite state Markov chains are extended to the nonlinear case. An operatorT inl 1 of a finite space with counting measure is called nonexpansive if ‖Tf-Tg‖1≦‖f-g‖1 holds for allf, g. It is shown that, for anyf, there exists an integer p>=1 such thatT pnf converges. Sufficient conditions forp=1 are given. In the case of continuous parameter nonexpansive semigrous {T t, t>=0},T tf converges fort→∞.
The main tool is a geometric theorem on isometriesS of compact subsets of the abovel 1: It is shown that any orbit underS is finite.
The exponential speed of convergence does not extend from the Markov chain case to nonlinearT.
References
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This research has been done during a visit of M.A.A. to the University of Göttingen. The principal results were announced in C.R. Math. Rep. Acad. Sci. Canada Vol. VIII, 1. Feb. 1986
The research of this author is supported in part by an N.S.E.R.C.-Grant
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Akcoglu, M.A., Krengel, U. Nonlinear models of diffusion on a finite space. Probab. Th. Rel. Fields 76, 411–420 (1987). https://doi.org/10.1007/BF00960065
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DOI: https://doi.org/10.1007/BF00960065
Keywords
- Markov Chain
- Stochastic Process
- Probability Theory
- Compact Subset
- Nonlinear Model