Metastability for a class of dynamical systems subject to small random perturbations.(English)Zbl 0709.60058

Summary: We consider dynamical systems in $${\mathbb{R}}^ d$$ driven by a vector field $$b(x)=-\nabla \alpha (x)$$, where $$\alpha$$ is a double-well potential with some smoothness conditions. We show that these dynamical systems when subjected to a small random disturbance exhibit metastable behavior in the sense defined by M. Cassandro and the authors [J. Stat. Phys. 35, 603-634 (1984; Zbl 0591.60080)].
More precisely, we prove that the process of moving averages along a path of such a system converges in law when properly normalized to a jump Markov process. The main tool for our analysis is the theory of M. I. Frejdlin and A. D. Venttsel’, Random perturbations of dynamical systems (1984; Zbl 0522.60055).

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes 60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 0591.60080; Zbl 0522.60055
Full Text: