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Ergodicity of diffusion and temporal uniformity of diffusion approximation

Published online by Cambridge University Press:  14 July 2016

M. Frank Norman
M. Frank Norman*
Affiliation:
University of Pennsylvania
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Abstract

Let {XN(t), t ≧ 0}, N = 1, 2, … be a sequence of continuous-parameter Markov processes, and let TN(t)f(x) = Ex[f(XN(t))]. Suppose that limN→∞TN(t)f(x)= T(t)f(x), and that convergence is uniform over x and over t ∈ [0, K] for all K < ∞. When is convergence uniform over t ∈ [0, ∞)? Questions of this type are considered under the auxiliary condition that T(t)f(x) converges uniformly over x as t → ∞. A criterion for such ergodicity is given for semigroups T(t) associated with one-dimensional diffusions. The theory is illustrated by applications to genetic models.

Keywords

DIFFUSION APPROXIMATIONERGODIC THEORYGENETIC MODELS
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 2 , June 1977 , pp. 399 - 404
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Ethier, S. N. (1975) An error estimate for the diffusion approximation in population genetics. Ph.D. Dissertation, University of Wisconsin.Google Scholar
[2] Ewens, W. J. (1969) Population Genetics. Methuen, London.CrossRefGoogle Scholar
[3] Freedman, D. (1971) Brownian Motion and Diffusion. Holden-Day, San Francisco.Google Scholar
[4] Kurtz, T. G. (1969) Extensions of Trotter's operator semigroup approximation theorems. J. Funct. Anal. 3, 354–375.CrossRefGoogle Scholar
[5] Mandl, P. (1968) Analytical Treatment of One-Dimensional Markov Processes. Springer-Verlag, New York.Google Scholar
[6] Maruyama, G. and Tanaka, H. (1957) Some properties of one-dimensional diffusion processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 11, 117–141.Google Scholar
[7] Norman, M. F. (1972) Markov Processes and Learning Models. Academic Press, New York.Google Scholar
[8] Norman, M. F. (1975) Diffusion approximation of non-Markovian processes. Ann. Prob. 3, 358–364.CrossRefGoogle Scholar