Weighted $L_\infty$ Asymptotic Characterisation of Perturbed Autonomous Linear Ordinary and Stochastic Differential Equations: Part I -- ODEs

TL;DR

This paper establishes necessary and sufficient conditions for the asymptotic behavior of solutions to perturbed linear differential equations, linking forcing function behavior with solution growth or decay, applicable to both scalar and multi-dimensional cases.

Contribution

It provides a weighted $L_inity$ asymptotic characterization of solutions to perturbed linear ODEs, extending to stochastic cases in Part II, with conditions on forcing functions for growth or decay.

Findings

Conditions for solution growth or decay based on forcing functions.

Extension of results to stochastic differential equations in Part II.

Applicability to scalar and multi-dimensional equations.

Abstract

This is the first of a two-part paper which determines necessary and sufficient conditions on the asymptotic behaviour of forcing functions so that the solutions of additively pertubed linear differential equations obey certain growth or decay estimates. Part I considers deterministic equations, and part II It\^o-type stochastic differential equations. Results from part I are used to deal with deterministically and stochastically forced equations in the second part. Results apply to both scalar and multi-dimensional equations, and connect the asymptotic behaviour of time averages of the forcing terms on finite intervals with the growth or decay rate of the solution. Mainly, results deal with large perturbations, but some indications of how results extend to tackle subdominant perturbations are also sketched.