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

TL;DR

This paper demonstrates that adding Brownian noise to a coupled complex-valued system of ODEs with finite-time blow-up solutions can stabilize the system and ensure ergodicity, extending previous results to higher dimensions.

Contribution

It extends the stabilization results from single complex-valued ODEs to coupled $ ext{C}^2$-valued systems using stochastic analysis techniques.

Findings

Brownian noise stabilizes the coupled system.

The system becomes ergodic after noise addition.

Analytical and numerical evidence supports stabilization.

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.