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2012 Propagating Lyapunov functions to prove noise-induced stabilization
Avanti Athreya, Tiffany Kolba, Jonathan Mattingly
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Electron. J. Probab. 17: 1-38 (2012). DOI: 10.1214/EJP.v17-2410


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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Avanti Athreya. Tiffany Kolba. Jonathan Mattingly. "Propagating Lyapunov functions to prove noise-induced stabilization." Electron. J. Probab. 17 1 - 38, 2012.


Accepted: 2 November 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1308.37003
MathSciNet: MR2994844
Digital Object Identifier: 10.1214/EJP.v17-2410

Primary: 37A25
Secondary: 37A30 , 37B25 , 60H10

Keywords: Invariant measures , Lyapunov functions , SDEs , Stochastic Stabilization


Vol.17 • 2012
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