2001/10/09 by Nils Berglund, Barbara Gentz
Environmental Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Ecosystem dynamics and resilience #cond-mat.dis-nn #cond-mat.stat-mech #math.DS #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/0305-4470/35/9/301
published as J. Phys. A 35:2057-2091 (2002) · 37 pages, 11 figures
arxiv created 2001/10/09 · openalex publication_date 2002/02/25 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
We introduce a new method, allowing one to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. In particular, scaling laws can be obtained easily. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.