Geometry's fundamental role in the stability of stochastic differential equations

Abstract

We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the “right” direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. In view of [Has80, JK85, Jur97, MT93b, RB06, SV72], the result is proven by using Lyapunov functions and geometric control theory.