Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-30T15:32:47.177Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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→∞ T N (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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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