Large fluctuations and growth rates of linear Volterra summation equations

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.

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