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Transactions of the American Mathematical Society

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Random evolution processes with feedback

Author: Kyle Siegrist
Journal: Trans. Amer. Math. Soc. 265 (1981), 375-392
MSC: Primary 60J25
MathSciNet review: 610955
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Abstract: A general random evolution Markov process is constructed which switches back and forth at random among a given collection of Markov processes ("modes of evolution") defined on a common evolution state space and indexed by an evolution rule space. Feedback is incorporated by allowing the path of the evolution component to influence the changes in evolution rule. The semigroup of the random evolution process is studied and is used to compare the process with the operator random evolutions of Griego and Hersh. Using deterministic modes of evolution, we generalize the Markov processes constructed by Erickson and by Heath. We also study new random evolution processes constructed from Brownian motions and from regular step processes.

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  • [1] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Academic Press, New York, 1968. MR 0264757 (41:9348)
  • [2] E. B. Dynkin, Markov processes (2 vols.), Springer-Verlag, Berlin, 1965.
  • [3] R. V. Erickson, Paths of random evolutions, Z. Wahrsch. Verw. Gebiete 29 (1974), 309-321. MR 0418252 (54:6293)
  • [4] R. J. Griego and R. Hersh, Random evolutions, Markov chains, and systems of partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 62 (1965), 305-308. MR 0270207 (42:5099)
  • [5] -, Theory of random evolutions with applications to partial differential equations, Trans. Amer. Math. Soc. 156 (1971), 405-418. MR 0275507 (43:1261)
  • [6] R. J. Griego and A. Moncayo, Random evolutions and piecing out of Markov processes, Bol. Soc. Mat. Mexicana (2) 15 (1970), 22-29. MR 0365723 (51:1975)
  • [7] G. J. Habetler and M. A. Martino, Existence theorems and spectral theory for the multigroup diffusion model, Proc. Sympos. Appl. Math., vol. 11, Amer. Math. Soc., Providence, R.I., 1959. MR 0144739 (26:2280)
  • [8] D. C. Heath, Probabilistic analysis of certain hyperbolic systems of partial differential equations, Ph.D. dissertation, Univ. Illinois, 1969.
  • [9] R. Hersh, Random evolutions: a survey of results and problems, Rocky Mountain J. Math. 4 (1974), 443-477. MR 0394877 (52:15676)
  • [10] N. Ikeda, M. Nagasawa and S. Watanabe, A construction of Markov processes by piecing out, Proc. Japan. Acad. Ser. A Math. Sci. 42 (1966), 370-375. MR 0202197 (34:2070)
  • [11] -, Branching Markov processes. I, J. Math. Kyoto Univ. 8 (1968), 233-278. MR 0232439 (38:764)
  • [12] R. P. Kertz, Limit theorems for discontinuous random evolutions with applications to initial value problems and to Markov processes on $ N$ lines, Ann. Probab. 2 (1974), 1046-1064. MR 0368180 (51:4421)
  • [13] -, Perturbed semigroup limit theorems with applications to discontinuous random evolutions, Trans. Amer. Math. Soc. 199 (1974), 29-53. MR 0362521 (50:14961)
  • [14] -, Random evolutions with underlying semi-Markov processes, Publ. Res. Inst. Math. Sci. 14 (1978), 589-614. MR 527191 (80m:60090)
  • [15] M. Pinsky, Multiplicative operator functionals and their asymptotic properties, Advances in Probability, III, Decker, New York, 1974. MR 0368182 (51:4423)
  • [16] K. T. Siegrist, Random evolutions with feedback, Ph.D. dissertation, Georgia Institute of Technology, 1979.

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Keywords: Markov process, random evolution, semigroup, infinitesimal operator, Brownian motion, regular step process
Article copyright: © Copyright 1981 American Mathematical Society

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