## The ODE method and spectral theory of Markov operators (2002)

Venue: | IN STOCHASTIC THEORY AND CONTROL (LAWRENCE, KS, 2001), SER. LECTURE NOTES IN CONTROL AND INFORMATION SCIENCES |

Citations: | 11 - 3 self |

### BibTeX

@INPROCEEDINGS{Huang02theode,

author = {Jianyi Huang and Ioannis Kontoyiannis and Sean P. Meyn},

title = {The ODE method and spectral theory of Markov operators},

booktitle = {IN STOCHASTIC THEORY AND CONTROL (LAWRENCE, KS, 2001), SER. LECTURE NOTES IN CONTROL AND INFORMATION SCIENCES},

year = {2002},

pages = {205--221},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a development of the ODE method for the analysis of recursive algorithms described by a stochastic recursion. With variability modeled via an underlying Markov process, and under general assumptions, the following results are obtained: (i) Stability of an associated ODE implies that the stochastic recursion is stable in a strong sense when a gain parameter is small. (ii) The range of gain-values is quantified through a spectral analysis of an associated linear operator, providing a non-local theory, even for nonlinear systems. (iii) A second-order analysis shows precisely how variability leads to sensitivity of the algorithm with respect to the gain parameter. All results are obtained within the natural operator-theoretic framework of geometrically ergodic Markov processes.