Noise-induced stabilization of planar flows I
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© 2015 University of Washington. All rights reserved.We show that the complex-valued ODE (n ≥ 1, an+1 6≠ 0): ź = an+1zn+1 + anzn +1zn + a0; which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant probability measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II  extends the main results to the general setting.
Published Version (Please cite this version)10.1214/EJP.v20-4047
Publication InfoHerzog, David P; & Mattingly, Jonathan Christopher (2015). Noise-induced stabilization of planar flows I. Electronic Journal of Probability, 20. 10.1214/EJP.v20-4047. Retrieved from https://hdl.handle.net/10161/10770.
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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