Noise-induced Stabilization of Planar Flows I
نویسندگان
چکیده
We show that the complex-valued ODE żt = an+1z n+1 + anz n + · · ·+ 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 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 [8] extends the main results to the general setting.
منابع مشابه
Noise-induced Stabilization of Planar Flows Ii
We continue the work started in Part I [6], showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Pe...
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