Propagating Lyapunov functions to prove noise-induced stabilization

Avanti Athreya (The Johns Hopkins University)
Tiffany Kolba (Valparaiso University)
Jonathan C Mattingly (Duke University)


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.

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Pages: 1-38

Publication Date: November 2, 2012

DOI: 10.1214/EJP.v17-2410


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