Propagating Lyapunov Functions to Prove Noise--induced Stabilization

TL;DR

This paper demonstrates how additive noise can stabilize a non-globally stable deterministic system in the plane by constructing a Lyapunov function through a systematic, meta-algorithmic approach, and verifies properties using Malliavin calculus.

Contribution

It introduces a systematic method for constructing Lyapunov functions via Poisson equations and presents a meta-algorithm applicable to noise-induced stabilization problems.

Findings

Noise induces a unique invariant measure with exponential convergence.

The Lyapunov function is constructed systematically using a sequence of Poisson equations.

Positivity of transition density is proved using explicit Malliavin calculus calculations.

Abstract

We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations.