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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 I 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. [ABSTRACT FROM AUTHOR]