The ODE Method and Spectral Theory of Markov Operators
  We give a development of the ODE method for the analysis of recursive algorithms described by a stochastic recursion. With variability modelled via an underlying Markov process, and under general assumptions, the following results are obtained: 1. Stability of an associated ODE implies that the stochastic recursion is stable in a strong sense when a gain parameter is small. 2. The range of gain-values is quantified through a spectral analysis of an associated linear operator, providing a non-local theory. 3. 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. 
J. Huang, I. Kontoyiannis, S.P. Meyn