Stabilization by Noise of a ${\mathbb{C}}^2$-Valued Coupled System

Abstract

Recently Herzog and Mattingly have shown that a $\mathbb{C}$-valued polynomial ODE which admits finite-time blow-up solutions may be stabilized by the addition of $\mathbb{C}$-valued Brownian noise. In this paper we extend their problem to a ${\mathbb{C}}^2$-valued system of coupled ODEs that also admits finite-time blow-up solutions. We show analytically and numerically that stabilization can be achieved in our setting by adding a suitable Brownian noise, and that the resulting system of SDEs is ergodic. The proof uses Girsanov theorem to effect a time change from our ${\mathbb{C}}^2$-system to a quasi-$\mathbb{C}$-system similar to the one studied by Herzog and Mattingly.

v4: 17 pages, 5 figures. Small typos corrected, refs added, section 1 revised. To appear in Stochastics and Dynamics

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