Abstract
We consider the rate of convergence to equilibrium of Volterra integrodifferential equations with infinite memory. We show that if the kernel of Volterra operator is regularly varying at infinity, and the initial history is regularly varying at minus infinity, then the rate of convergence to the equilibrium is regularly varying at infinity, and the exact pointwise rate of convergence can be determined in terms of the rate of decay of the kernel and the rate of growth of the initial history. The result is considered both for a linear Volterra integrodifferential equation as well as for the delay logistic equation from population biology.
Highlights
In this paper we consider the asymptotic behaviour of linear and nonlinear Volterra integrodifferential equations with infinite memory, paying particular attention to the connection between the asymptotic behaviour of the initial history as t → −∞ and the rate of convergence of the solution to a limit
The process can have a limiting autocovariance function which may differ from that of the stationary process. This phenomenon is impossible for processes with bounded memory, and the different convergence rates which depend on the asymptotic behaviour of the initial history in this case is an exact analogue to the history-dependent decay rates recorded here
Once Theorem 3.4 has been proven, we are able to determine the rate of decay of the solution of the following linear infinite memory convolution equation t x t ax t b t − s x s ds, t > 0
Summary
In this paper we consider the asymptotic behaviour of linear and nonlinear Volterra integrodifferential equations with infinite memory, paying particular attention to the connection between the asymptotic behaviour of the initial history as t → −∞ and the rate of convergence of the solution to a limit. In the case of the so-called ARCH ∞ processes which are stationary, the resulting equation for the autocovariance function of the process can be represented as a Volterra summation equation with infinite memory. The process can have a limiting autocovariance function which may differ from that of the stationary process This phenomenon is impossible for processes with bounded memory, and the different convergence rates which depend on the asymptotic behaviour of the initial history in this case is an exact analogue to the history-dependent decay rates recorded here. An up-to-date survey of work on long memory processes is given by Cont 11
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