Stabilization and destabilization of nonlinear differential equations by noise

This paper considers the stabilization and destabilization by a Brownian noise perturbation that preserves the equilibrium of the ordinary differential equation x'(t) = f (x (t)). In an extension of earlier work, we lift the restriction that f obeys a global linear bound, and show that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g (X (t)) dB (t) either stabilizes an unstable equilibrium, or destabilizes a stable equilibrium. When the equilibrium of the deterministic equation is nonhyperbolic, we show that a nonhyperbolic perturbation suffices to change the stability properties of the solution.