Noise-Induced Stabilization of Planar Flows I

TL;DR

This paper demonstrates that adding a small stochastic noise to certain complex-valued differential equations can stabilize their trajectories, leading to a unique, heavy-tailed invariant measure, with the approach based on Lyapunov functions.

Contribution

It introduces a novel Lyapunov function-based method to show noise-induced stabilization of planar flows, applicable to complex polynomial ODEs.

Findings

Stochastic noise stabilizes finite-time blow-up trajectories.

The invariant measure is heavy-tailed and exponentially attracting.

Methodology is general and adaptable to other systems.

Abstract

We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} 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 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 of this paper extends the main results to the general setting.