Large fluctuations and growth rates of linear Volterra summation equations

TL;DR

This paper analyzes the asymptotic behavior of solutions to linear Volterra summation equations with unbounded forcing, revealing how growth and fluctuations in the forcing influence the solutions, with extensions to stochastic and nonlinear cases.

Contribution

It provides new asymptotic representations for solutions under various forcing growth conditions and extends the analysis to stochastic and nonlinear equations.

Findings

Solutions mirror the growth or fluctuations of the forcing term.

Asymptotic growth bounds are established for solutions.

The theory applies to both deterministic and stochastic equations.

Abstract

This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples. Finally, we show that our theory can be extended to cover a class of nonlinear equations via a straightforward linearisation argument.