Subexponential solutions of linear integro-differential equations and transient renewal equations

Abstract

This paper studies the asymptotic behaviour of the solutions of the scalar integro-di® erential equationThe kernel k is assumed to be positive, continuous and integrable. Ifit is known that all solutions x are integrable and x(t) ! 0 as t ! 1 , but also that x = 0 cannot be exponentially asymptotically stable unless there is some ® > 0 such that Z 1 0 k(s)e ® s ds < 1 :Here, we restrict the kernel to be in a class of subexponential functions in which k(t) ! 0 as t ! 1 so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) ! 0 as t ! 1 is given byThe result is proved by determining the asymptotic behaviour of the solution of the transient renewal equationIf the kernel h is subexponential, then lim t! 1 r(t) h(t) = 1 (1 ¡ R 1 0 h(s) ds) 2 :

Almost sure subexponential decay rates of scalar Ito-Volterra equations

“…This result has an important corollary for solutions of (3). When a > ∞ 0 k(s) ds, and k decays to zero polynomially according to lim t→∞ log k(t) log t = −α, for some α > 1, then almost sure decay rate of the solution as the noise intensity becomes arbitrarily large is approximately tk(t), in the sense that lim sup t→∞ log |X σ (t)| log(tk(t)) = 1 − Λ(|σ|), a.s.…”

“…Lemma 4.3 in [3] further enables us to conclude that X is positive subexponential, so (ii) of Theorem 10 and Theorem 11 follow. By Lemma 14, we know that part (i) of these Theorems also hold.…”

“…In order to state it, we first recall the definition of a subexponential function, introduced in [3].…”

“…A discussion of this class in contained in [3]. Note however that it contains for example, all positive and integrable functions which are regularly varying at infinity, as well as functions positive functions which obey k(t) ∼ Ce −t α , as t → ∞, for some C > 0 and α ∈ (0, 1).…”

Mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equations

“…Hence if it fails to hold, a uniformly asymptotically stable solution cannot be exponentially asymptotically stable. Some deeper related work on deterministic equations by Appleby can be found in [10][11][12], including the so-called "non-exponential decay rate" and "subexponential solution". Mao [13] investigated the mean square stability of the generalized equation…”