Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach

Nils Berglund (1) , Barbara Gentz (2)

1 CPT - Centre de Physique Théorique - UMR 6207
2 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

Stochastic differential equations play an increasingly important role in modeling the dynamics of a large variety of systems in the natural sciences, and in technological applications. This book is aimed at advanced undergraduate and graduate students, and researchers in mathematics, physics, the natural sciences, and engineering. It presents a new constructive approach to the quantitative description of solutions to systems of stochastic differential equations evolving on well-separated timescales. The method, which combines techniques from stochastic analysis and singular perturbation theory, allows the domains of concentration for typical sample paths to be determined, and provides precise estimates on the transition probabilities between these domains. In addition to the detailed presentation of the set-up and mathematical results, applications to problems in physics, biology, and climatology are discussed. The emphasis lies on noise-induced phenomena such as stochastic resonance, hysteresis, excitability, and the reduction of bifurcation delay.

Mots clés: Slow-fast dynamical systems; Singular perturbation theory; Dynamic bifurcations; Stochastic differential equations; Noise; First-exit time; Sample paths; Large deviations; Concentration of paths; Stochastic resonance; Hysteresis; Climate models; Neural models

Domaines: Probabilités [math.PR]

Nils Berglund, Barbara Gentz. Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach. Springer, Springer, Probability and its Applications, 290 p., 2005, 978-1-84628-038-2. ⟨10.1007/1-84628-186-5⟩. ⟨hal-00010168⟩