Dynamical systems perturbed by big jumps: meta-stability and non-local search

Ilya Pavlyukevich

We consider a dynamical system driven by a vector field -U´, where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a symmetric a-stable Lévy process, or more generally by a Lévy process with heavy tails. We show that the perturbed dynamical system exhibits metastable behavior, i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature of the random perturbation, the transition times and probabilities differ strongly from the well-known Gaussian case.
Further, making the noise amplitude decreasing with the time, we study the Lévy-driven jump-diffusions in the simulated annealing regime, and consider its applications to non-local search in multi-dimensional landscapes and to stochastic optimisation.

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