Math @ Duke

Publications [#243875] of Jonathan C. Mattingly
Papers Published
 with Athreyaz, A; Kolba, T; Mattingly, JC, Propagating lyapunov functions to prove noiseinduced stabilization,
Electronic Journal of Probability, vol. 17
(November 2, 2012),
pp. 138, ISSN 10836489 [math/111.1755], [repository], [doi]
(last updated on 2017/12/12)
Abstract: We investigate an example of noiseinduced 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 metaalgorithm 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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