Electronic Journal of Probability

Propagating Lyapunov functions to prove noise-induced stabilization

Avanti Athreya, Tiffany Kolba, and Jonathan Mattingly

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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.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 96, 38 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 37B25: Lyapunov functions and stability; attractors, repellers 60H10: Stochastic ordinary differential equations [See also 34F05]

SDEs Lyapunov Functions Invariant Measures Stochastic Stabilization

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Athreya, Avanti; Kolba, Tiffany; Mattingly, Jonathan. Propagating Lyapunov functions to prove noise-induced stabilization. Electron. J. Probab. 17 (2012), paper no. 96, 38 pp. doi:10.1214/EJP.v17-2410. https://projecteuclid.org/euclid.ejp/1465062418

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