Growth and fluctuation in perturbed nonlinear Volterra equations

TL;DR

This paper establishes precise bounds on the growth and fluctuation of solutions to nonlinear Volterra equations under external perturbations, extending results to stochastic cases with Brownian and Lévy noise.

Contribution

It provides a detailed analysis of the asymptotic behavior of solutions to perturbed nonlinear Volterra equations, including stochastic variants, based on the limiting behavior of the forcing term.

Findings

Solutions' growth rates depend on the limit of the forcing term at infinity.

Asymptotic behavior varies with the limit L: like unforced when L=0, like forcing when L=+∞.

Results extend naturally to stochastic equations with additive noise.

Abstract

We develop precise bounds on the growth rates and fluctuation sizes of unbounded solutions of deterministic and stochastic nonlinear Volterra equations perturbed by external forces. The equation is sublinear for large values of the state, in the sense that the state--dependence is negligible relative to linear functions. If an appropriate functional of the forcing term has a limit $L$ at infinity, the solution of the differential equation behaves asymptotically like the underlying unforced equation when $L=0$, like the forcing term when $L=+\infty$, and inherits properties of both the forcing term and underlying differential equation for values of $L\in (0,\infty)$. Our approach carries over in a natural way to stochastic equations with additive noise and we treat the illustrative cases of Brownian and L\'evy noise.