Noise-induced stabilization of planar flows I

Abstract

We show that the complex-valued ODE ($n\geq 1$, $a_{n+1} \neq 0$): $$\dot z = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0 , $$ which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant probability measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II extends the main results to the general setting.