2003/02/02 by Natella V. O'Bryant
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:37J40 #msc:60F17
paper · pdf · doi:10.1142/s0219493703000668
published as Stochastics and Dynamics, Vol. 3, No. 1 (2003) 1-54 · 53 pages, 2 figures
arxiv created 2003/02/02 · openalex publication_date 2003/03/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a two-dimensional weakly dissipative dynamical system with time-periodic drift and diffusion coefficients. The average of the drift is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. Using the martingale problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficients, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of time scales, are similar to the conditions on the domain of skew Brownian motion and are related to the description of spider martingales.