FUNCTIONAL DIFFERENTIAL EQUATIONS VOLUME 11 24, NO 1-2 PP. 5 10 SUBEXPONENTIAL SOLUTIONS OF LINEAR ITO-VOLTERRA EQUATIONS WITH A DAMPED PERTURBATION J. A. D. APPLEBY' Abstract. This paper studies the almost sure non-exponential decay rate of solutions of a scalar linear Ito-Volterra equation with state-independent diffusion coefficient. If the kernel is subexponential, and the diffusion term decays sufficiently quickly, the decay rate is subexponential, and the same as in the deterministic case. If the diffusion coefficient is subexponential, there is a subexponential upper bound on the decay rate of solutions. Key Words. Volterra integra-differential equations, exponential asymptotic stability, subexponential functions. AMS(MOS) subject classification. 34K20, 34K50, 60Hl0 1. Introduction. In this paper we study the asymptotic behaviour of the scalar linear convolution Ito-Volterra equation (1) dx(t) = (-ax(t) + (k * X)(t)) dt +!J(t) db(t), where k is continuous, positive and integrable on [0, oo), and (J is continuous on [0, oo). Here, f * g denotes the convolution off, g E C(O, oo) (f * g)(t) = t f(t- s)g(s) ds. We assume that the initial condition X(O) = ~ is deterministic, and, as conventional, that (1) is shorthand for the integral equation (2) X(t)=~+ { 1 {(-ax(s)+(bx)(s)}ds+ f'(j(s)db(s), t;::o:o. lo.~ Centre for Modelling with Differential Equations (CMDE), School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland 5
6 J. A. D. APPLEBY The probabilistic setting for this equation is a complete filtered probability space (fj, F, (F(t)Jt?:o, P), where B is a scalar standard Brownian motion. According to [7], there is a unique continuous process, adapted to the filtration, which is a strong solution of (1). In the following, the abbreviation is used for "almost sure" or "almost surely"; in each case these refer to almost sure events relative to the objective probability measure P. Let z be the unique solution of (3) z'(t) = -az(t) + (k * z)(t), t 2 0, z(o) = 1. In [4], the following necessary and sufficient conditions for the exponential stability of solutions of (1) were established: (a) The solution of ( 3) is integrable and (4) fa'"' k(s)e 7 '' ds < oo for some 1 1 > 0, (5) fooo e 27 "a(s? ds < oo for some 12 > 0. (b) There is /3 0 > 0 such that all solutions of (1) obey limsup~logix(t)l :<:: -/3o, t->oo t Therefore, solutions cannot decay to zero exponentially if k or a do not obey the exponential decay criteria (4), (5). Thus, when solutions of (1) are asymptotically stable (see, e.g., [1]), they obey limsup~logix(t)l = 0, t->oo t In this paper, we find precise estimates on the non-exponential decay of solutions, under non-exponential hypotheses on k and a. Similar hypotheses are used to study non-exponential stability in Ito-Volterra equations in [6, 3]. 2. Subexponential functions and deterministic equations. The following class of functions, introduced in [5], derives from a definition in [8]. DEFINITION 1. We say k E 1 (R+) n C(R+; R") is subexponential if. (k * k)(t) (6) I 1m -'-~-'-'- Hoo k(t) (7) lim sup \k(t- s) - 1\ t->oo se[o,r] k(t) 0, \1 T > 0.
SUBEXPONENTIAL STOCHASTIC VOLfERRA EQUATIONS 7 If k obeys these properties, we write k E U. As pointed out in [5], condition (7) implies that k obeys (8) lim k(t)e t = oo, for all c > 0. t->oo Thus, if k is subexponential, it obeys neither (4) nor (5). The class of subexponential functions is discussed in detail in [5]. It contains, for example, all positive functions which are regularly varying at infinity, with index a < -1. Denote by BC the space of bounded continuous functions on (0, oo), and BCh the space of continuous functions f with f /he BC. For f E BCh, let. lf(t) I Ahf = lim sup -h( ). t-+oo t As in [5], we denote by BCf, the space of continuous functions f E BCh where lim,_, $t exists. For f E BCh, we write. f(t) Lhf = lim h( ). t-+oo t We will frequently use the following lemma, proved in [5]. LEMMA 1. LethE U. Iff, g E BC}u then f * g E BC~ and Lh(f *g)= Lhf fooo g(s) ds + Lhg fooo f(s) ds. In this paper, we state an analogous result for functions in BCh As it may be proved in a similar fashion to Lemma 1, its proof is omitted. LEMMA 2. LethE U. Iff, g E BC 1 " then f * g E BCh and Ah(lfl * lgl) :S Ahf 1 lg(s)l ds + Ahg 1 lf(s)l ds. These results yield the following perturbation theorem, partly proven in [5]. THEOREM 1. Let f E C(O,oo) n L 1 (0,oo), k E C([O,oo);(O,oo)) n 1(0, oo), a>.f 0 k(s) ds, and x be the solution of (9) x'(t) = -ax(t) + (k * x)(t) + f(t), t?: 0. (a) rr k E u, f E BCL then Lkx = Lkf +.f0 x(s) ds a- /~ k(s)ds (b) Iff obeys (6), ('l), and Ltk = 0, then Lfx =(a-.f 0 k(s) dsr 1. This result informs our study of the stochastically perturbed equation (1). It suggests that we find a deterministic proxy for the size of the random perturbation, to determine the critical decay rate of a at which there is a transition from "small" to "large perturbation" asymptotics.
8 J. A. D. APPLEBY 3. Asymptotic stability of (1). We now study the asymptotic stability of solutions of (1) where a E 2 (0, oo). This allows for direct comparison with equation (9) when f E U(O,oo), for when a E L 2 (0,oo), the Ito integral on the righthand side of (2) has a finite limit as t -+ oo, THEOREM 2. If the solution of (3} is in U(O,oo), k E C((O,oo)) n U(O, oo), and a E C(O, oo) n 2 (0, oo), then (a) limt->ooe[x(t) 2 ] = 0, E[X 2 ] E U(O,oo), and {b)!imt->oox(t) = 0, X E L 2 (0,oo), Proof. With z defined by (3), the solution of (1) is X(t) = X(O)z(t) + Y(t), where Y(t) = JJ z(t- s)a(s) db(s) (see [1, 9]). In [1] it is noted that Y(t) is a normally distributed random variable with zero mean and variance v(t) 2, where v(t) 2 = (z 2 * a 2 )(t). Since z E 1 (0, oo) implies z(t) -+ 0 as t-+ oo, z 2 E U(O,oo). But a 2 E L 1 (0,oo) gives v(t)-+ 0 as t-+ oo and v E 2 (0, oo). Since E[X(t) 2 ] ::; 2z(tj2 X(Oj2 + 2v(t) 2, part (a) holds. Part (b) follows by part (a) and the method of proof of Theorem 1 in [2]. D Ask is positive here, the condition a> frf" k(s) ds implies z E L 1 (0, oo). 4. Main Results. The proofs of the main results in this paper rely on rewriting the solution of (1) in terms of the solution of a perturbed Volterra integrodifferential equation whose solution, although random, is in C 1 (0, oo). LEMMA 3. If k E C([O, oo); (0, oo)) n L 1 (0, oo), a> f 0 "" k(s) ds, then for a.a. we 0, the path X(w) obeys (10) X(t,w) = U(t,w) +T(t,w) where T(w) is the function defined by (11) T(t,w) =- (],"" a(s) db(s)) (w), and f(w), U(w) obey (12) (13) f(t, w) U'(t, w) -at(t,w) + (k * T(w))(t), -au(t,w) + (k * U(w))(t) + f(t,w). We refer the reader to a similar result, proven in [3], where a more general result on the asymptotic behaviour of the random variable T(t) also appears. LEMMA 4. Let a E C([O, oo); (0, oo)) n 2 (0, oo), and (14) I:;(t) 2 = 1"" a(s) 2 dsloglog ([Y" a(s) 2 ds) - 1.
SUBEXPONENTIAL STOCHASTIC VOLTERRA EQUATIONS 9 Then T defined by ( 11) obeys (15) I 1m. sup IT(t)l ( ) - _ v ;c; "'' 2 Hoo 2:: t This, along with Lemmas 1 and 2 suggests that the random perturbation f in (13) decays at the slower rate between k and E. Viewing Theorem 1 in the light of Lemma 3, the case in which LkE = 0 seems to correspond to part (a) of the theorem, while that in which L-;:;k = 0 roughly corresponds to part (b). The following result thus parallels Theorem 1, part (a). THEOREM 3. If k E C([O, oo); (0, oo)) n U(O, oo), u E L 2 (0, oo), u E C([O,oo); (O,oo)), a> f 0 k(s)ds, k is.sv.bexponential, and' obeys Lk' = 0, then the unique strong solution of (1) obeys (16) where G is a normally distributed F -measurable random variable, which is nonzero, and has mean U (a-f 0 k(s) ds), and variance.f(f" u(s ) 2 ds/ (af0 k(.s) ds) 2. Moreover, X(w) obeys (6), (7) for almost all we D. The conclusion of the first part of the result is precisely that of Theorem 1, when Ld = 0. So, although the sample paths of X are nowhere differentiable,, they behave asymptotically like k, which can be in C (0, oo). Proof. The second statement of the theorem follows from the first by arguments of [5] applied to each w in the set on which (16) holds, and G is non-trivial. As to the proof of the first part, the hypotheses and (15) imply LkT = 0 Since k is integrable, T E L 1 (0, oo), Applying Lemma 1 to fin (12) yields Lkf = j~ T(s) ds,, so by Theorem 1, U obeys L U= Lkf+j~ U(s)ds k a-j~ k(s)ds (10) now gives (16). To find the distribution of G, note that k E L 1 (0,oo) implies X E L 1 (0, oo). Therefore, letting t--+ oo on both sides of (2), and noting that u E L2(0, oo) implies limt->oo.f~ u(s) db(s) exists, gives {oo X(s) ds = ( + fooo o(s) db(s) lo a-j~ k(s)ds Since J(f" u(s) db(s) is normally distributed with variance J'(f" u(s) 2 ds, G has the claimed distribution. Since u '/= 0, G is not trivial, so LkX f 0, 0 We now supply the stochastic analogue of case (b) in Theorem 1.
10 J. A. D. APPLEBY THEOREM 4. If k E C([O,oo); (O,oo)) n l(o,oo), J E L 2 (0,oo) n C([O, oo); (0, oo)), a > f 0 k(s) ds, :E defined by {14) is subexponential, and Ar;k = L E [O,oo), then the unique strong solution of {1) obeys Ar;X < oo, Moreover, if Lr;k = 0 then 2v'2a Ar;X::; a- fooo k(s) ds' Proof We study the case where Lr;k = 0 only: the case when L =J 0 is similar and hence omitted. By (15), Ar;T = J2, Thus Lemma 2, Lr;k = 0, and (12) gives (17) A 2 ;/::; v'2(a + f' k(s) ds), Define ea(t) =e-at, h = ea*k: then z = ea+ea*t, where r solves r = h+h*r. Then Lr;k = 0, Lemma 2 imply Ar;h = 0. An argument involving Lemma 2 yields Lr;r = 0, so Lz;z = Ar;z = 0. As Ar;z = 0, Ar;U = Ar;(z *f). The result now follows, by Lemma 2, (13), (15),and (17). 0 Acknowledgements The author is very grateful for discussions with Kieran Murphy, and also to the anonymous referee for their careful reading of the manuscript. REFERENCES [1] J. A. D. Appleby, Almost sure asymptotic stability of linear Ito-Volterra equations with damped stochastic perturbations, Electron. Comm. Probab., 7(22), 223-234. [2] J. A. D. Appleby, p'h mean integrability and almost sure asymptotic stability of Ito-Volterra equations, J. Integral Equations Appl., (23), to appear. [3] J. A. D. Appleby, Almost sure subexponential decay rates of scalar Ito-Volterra equations, Electron. J. Qual. Theory Differ. Equ., (23), to appear. [4] J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear Ito Volterra equations with damped stochastic perturbations, Electron. J. Probab., 23, to appear. [5] J. A. D. Appleby and D. W. Reynolds, Subexponential solutions of linear Volterra integra-differential equations and transient renewal equations, Proc. Roy. Soc. Edinburgh. Sect. A, 132A(22), 521-543. [6] J. A. D. Appleby and D. W. Reynolds, Non-exponential stability of scalar stochastic Volterra equations, Statist. Probab. Lett., 62(4)(23), 335-343. [7] M. A. Berger and V. J. Mizel, Volterra equations with Ito integrals I, J. Integral Equations, 2(3)(1980), 187-245. [8] J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Analyse. Math., 26(1972), 255-302. [9] U. Kuchler and S. Mensch, Langevin's stochastic differential equation extended by a time-delay term, Stochastics Stochastics Rep., 40(1-2)(1992), 23-42.