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

TL;DR

This paper establishes precise conditions under which the convergence rates of solutions to differential equations with regularly varying nonlinearities are preserved under deterministic and stochastic perturbations, including finite difference approximations.

Contribution

It provides necessary and sufficient conditions for preserving asymptotic convergence rates and derivative behaviors in perturbed differential equations, including stochastic cases.

Findings

Sharp conditions for convergence rate preservation

Conditions for derivative behavior preservation

Finite difference approximation conditions

Abstract

This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to their equilibrium. Sharp conditions are also established which preserve the asymptotic behaviour of the derivative of the underlying unperturbed equation. Finally, necessary and sufficient conditions are established which enable finite difference approximations to the derivative in the stochastic equation to preserve the asymptotic behaviour of the derivative of the unperturbed equation, even though the solution of the stochastic equation is nowhere differentiable, almost surely.