On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity

摘要

This paper develops necessary and sufficient conditions for the preservationof asymptotic convergence rates of deterministically and stochasticallyperturbed ordinary differential equations with regularly varying nonlinearityclose to their equilibrium. Sharp conditions are also established whichpreserve the asymptotic behaviour of the derivative of the underlyingunperturbed equation. Finally, necessary and sufficient conditions areestablished which enable finite difference approximations to the derivative inthe stochastic equation to preserve the asymptotic behaviour of the derivativeof the unperturbed equation, even though the solution of the stochasticequation is nowhere differentiable, almost surely.