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Finite regular invariant measures for Feller processes

Published online by Cambridge University Press:  14 July 2016

V. E. Beneš
V. E. Beneš*
Affiliation:
Bell Telephone Laboratories, Murray Hill, New Jersey
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Abstract

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 1 , April 1968, pp. 203 - 209
Copyright
Copyright © Sheffield: Applied Probability Trust 

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