Almost sure convergence of solutions of linear stochastic Volterra equations to nonequilibrium limits. (English) Zbl 1154.45012

Let \(B\) be a \(d\)-dimensional Brownian motion defined over some probability space \((\Omega,{\mathcal F},P)\), and let \(A\) be an \((n\times n)\)-matrix and \(K\) and \(\Sigma\) continuous matrix valued functions defined over \(\mathbb{R}_+\), where \(K\) is supposed to be integrable. The authors of the present paper investigate the asymptotic behavior of the solution \(X_t\), as \(t\rightarrow +\infty\), of the Volterra equation \[ dX_t=(AX_t+\int_0^tK(t-s)X_sds)\,dt+\Sigma_tdB_t,\;t\geq 0,\;X_0=x\in \mathbb{R}^n. \] While several authors studied the asymptotic convergence of solutions of this equation to the trivial solution, the authors of the present paper have been the first to study in an earlier work [Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13B, suppl., 515–534 (2006)] the convergence of \(X_t\) to a nonequilibrium point. Contrarily to their former paper in which the convergence was studied in \(L^2(\Omega)\)-sense, the present paper discusses the case of \(P\)-almost sure convergence of \(X_t\) to an explicit nonequilibrium point \(X_\infty.\) The authors prove that the sufficient conditions for convergence and integrability on the resolvent, the kernel and the noise in the \(L^2(P)\) case are also sufficient for the \(P\)-a.s. case. The conditions on the kernel and the resolvent are also shown to be necessary. Necessary and sufficient conditions for the \(P\)-a.s. convergence are discussed for the scalar case. Finally, the authors apply their results to an epidemiological Volterra model.


45R05 Random integral equations
45J05 Integro-ordinary differential equations
60H20 Stochastic integral equations
45M05 Asymptotics of solutions to integral equations
45M10 Stability theory for integral equations
Full Text: DOI


[1] J.A.D. Appleby, S. Devin and D.W. Reynolds, Mean square convergence of solutions of linear stochastic Volterra equations to nonequilibrium limits , Dynam. Contin. Discrete Impuls. Syst. Math. Anal., Page 439, first reference. A Math. Anal. 13B (2006), suppl., 515-534.
[2] J.A.D. Appleby and M. Riedle, Almost sure asymptotic stability of Volterra equations with fading perturbations , Stoch. Anal. Appl. 24 (2006), 813-826. · Zbl 1121.60070
[3] M.A. Berger and V.J. Mizel, Volterra equations with Itô integrals , I.J. Integral Equations 2 (1980), 187-245. · Zbl 0442.60064
[4] H. Brunner, A. Pedas and G. Vainikko, The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations , Math. Comp. 68 (1999), 1079-1095. JSTOR: · Zbl 0941.65136
[5] ——–, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels , SIAM J. Numer. Anal. 39 (2001), 957-982. · Zbl 0998.65134
[6] T.A. Burton, Fixed points and differential equations with asymptotically constant or periodic solutions , Electronic J. Qual. Theory Differential Equations 11 (2004), electronic. · Zbl 1083.34055
[7] K.L. Cooke and J.A. Yorke, Some equations modelling growth processes and gonorrhea epidemics , Math. Biosci. 16 (1973), 75-101. · Zbl 0251.92011
[8] G. Gripenberg, S.O. Londen and O. Staffans, Volterra integral and functional equations , Cambridge University Press, Cambridge, 199-0. · Zbl 0695.45002
[9] J. Haddock and J. Terjéki, On the location of positive limit sets for autonomous functional-differential equations with infinite delay , J. Differential Equations 86 (1990), 1-32. · Zbl 0725.34080
[10] T. Krisztin and J. Terjéki, On the rate of convergence of solutions of linear Volterra equations , Boll. Un. Mat. Ital. 2 (1988), 427-444. · Zbl 0648.45003
[11] X. Mao, Stability of stochastic integro-differential equations , Stochastic Anal. Appl. 18 (2000), 1005-1017. · Zbl 0969.60068
[12] X. Mao and M. Riedle, Mean square stability of stochastic Volterra integrodifferential equations , Systems Control Letters 55 (2006), 459-465. · Zbl 1129.34332
[13] R.K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels , SIAM J. Math. Anal. 2 (1971), 242-258. · Zbl 0217.15602
[14] L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales , Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 200-0. · Zbl 0977.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.