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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.

MSC:

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
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