Propagating lyapunov functions to prove noise-induced stabilization
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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.
Published Version (Please cite this version)10.1214/EJP.v17-2410
Publication InfoAthreyaz, A; Kolba, T; & Mattingly, JC (2012). Propagating lyapunov functions to prove noise-induced stabilization. Electronic Journal of Probability, 17. 10.1214/EJP.v17-2410. Retrieved from https://hdl.handle.net/10161/9518.
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James B. Duke Distinguished Professor
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day. He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and