This paper considers the stabilisation and destabilisation by a Brownian noise perturbation which 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 stabilises an unstable equilibrium, or destabilises a stable equilibrium. When the equilibrium of the deterministic equation is non-hyperbolic, we show that a non-hyperbolic perturbation suffices to change the stability properties of the solution.

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We consider stochastic difference equationwhere functions f and g are nonlinear and bounded, random variables ξ i are independent and h > 0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x n ≡ 0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x n is approximately polynomial: we find α > 0 such that x n decays faster than n −α+ε but slower than n −α−ε for any ε > 0.It also turns out that if g(x) decays faster than f (x) as x → 0, the polynomial rate of decay can be established exactly, x n n α → const. On the other hand, if the coefficient by the noise does not decay fast enough, the approximate decay rate is the best possible result.1991 Mathematics Subject Classification. 39A10, 39A11, 37H10, 34F05, 93E15.

This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in p-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in p-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in p-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.

We develop necessary and sufficient conditions for the a.s. asymptotic stability of solutions of a scalar, non-linear stochastic equation with state-independent stochastic perturbations that fade in intensity. These conditions are formulated in terms of the intensity function: roughly speaking, we show that as long as the perturbations fade quicker than some identifiable critical rate, the stability of the underlying deterministic equation is unaffected. These results improve on those of Chan and Williams; for example, we remove the monotonicity requirement on the drift coefficient and relax it on the intensity of the stochastic perturbation. We also employ different analytic techniques.