Stabilization by noise of a ℂ2-valued coupled system

Abstract

D. Herzog and J. Mattingly have shown that a [Formula: see text]-valued polynomial ODE with finite-time blow-up solutions may be stabilized by the addition of [Formula: see text]-valued Brownian noise. In this paper, we extend their results to a [Formula: see text]-valued system of coupled ODEs with 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 the Girsanov theorem to induce a time change from our [Formula: see text]-system to a quasi-[Formula: see text]-system similar to the one studied by Herzog and Mattingly.