First Exit Times of Non-linear Dynamical Systems in ℝ d Perturbed by Multifractal Lévy Noise

Abstract

In a domain \(\mathcal{G}\subset\mathbb{R}^{d}\) we study a dynamical system which is perturbed in finitely many directions i by one-dimensional Levy processes with α i -stable components. We investigate the exit behavior of the system from the domain in the small noise limit. Using probabilistic estimates on the Laplace transform of the exit time we show that it is exponentially distributed with a parameter that depends on the smallest α i . Finally we prove that the system exits from the domain in the direction of the process with the smallest α i .