Noise-induced stabilization of planar flows I

Abstract

We show that the complex-valued ODE (n ≥ 1, a n+1 = 0): z = a n+1 z n+1 + a n z n + · · · + a 0 , 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 [11] extends the main results to the general setting.

Stabilization by noise of a ℂ2-valued coupled system

“…Observe that, without the term of order e −2νt , if ν = 0, then (3.2) is a special case of the C-valued SDE with polynomial drift (1.1), studied in [9]. We now state several related results from [9] which will be used in the sequel [10].…”

“…the stochastic stabilization of which was studied by Herzog and Mattingly [9]. So on a heuristic level, our stabilization problem in C 2 , can be reduced to the stabilization problem in C,…”

“…The approach of [9] is to construct a Lyapunov function pair corresponding to the Itô process z t ; see Theorem 5.2 of [9] for the precise statement, and Secs. 6 and 7 of [9] for the proofs. Once this Lyapunov pair is constructed, then one can apply the general results from Sec.…”

“…In [3,4,7], the flow of certain fluids can be modeled by an SDE with a polynomial drift term. This inspired Herzog and Mattingly [8,9] to study the stability of the complex-valued SDE dz t = (a n+1 z n+1 t + a n z n t + · · · + a 0 )dt + σ dB t (1.1) with initial condition z 0 ∈ C, where B t = B (1) t + iB (2) t , B…”

“…We would like to extend the stability condition established in [8][9][10] to the stochastics Burgers' equation. To do so, we begin with a simple model of the Burger's equation stated in [11].…”