Conditional Averaging for Diffusive Fast-Slow Systems: A Sketch for Derivation

Summary This article is concerned with stochastic differential equations with disparate temporal scales. We consider cases where the fast mode of the system rarely switches from one almost invariant set in its state space to another one such that the time scale of the switching is as slow as the slow modes of the system. In such cases descriptions for the effective dynamics cannot be derived by means of standard averaging schemes. Instead a generalization of averaging, called conditional averaging, allows to describe the effective dynamics appropriately. The basic idea of conditional averaging is that the fast process can be decomposed into several ‘almost irreducible’ sub-processes, each of which can be treated by standard averaging and corresponds to one metastable or almost invariant state. Rare transitions between these states are taken into account by means of an appropriate Markov jump process that describes the transitions between the states. The article gives a derivation of conditional averaging for a class of systems where the fast process is a diffusion in a double well potential.