Behavior of solution of stochastic difference equation with continuous time under additive fading noise (Q2064498)

Consider the following difference equation on \(\mathbb{R}^n\) with continuous time and under the influence of an additive stochastic perturbation: \[ \begin{multlined} x(t+\tau)=a_1(t,x(t),x(t-h_1),x(t-h_2),\dots)\\ +a_2(t,x(t),x(t-h_1),x(t-h_2),\dots)\xi(t+\tau) +b(t)\eta(t+\tau), \end{multlined} \] with \( t>t_0-\tau\) and \[ x(\theta)=\phi(\theta),\quad \theta\in \Theta=[t_0-h,t_0],\quad h=\tau+\sup_{j\geq 1}h_j, \] where the functionals \(a_1\) and \(a_2\) are \(\mathbb{R}^n\)-valued and \(\mathbb{R}^{n\times k}\)-valued respectively, \(\tau,h_1,\dots\) are positive constants, \(\xi\) is a certain \(\mathbb{R}^k\)-valued stationary stochastic process, \(b(t)\in \mathbb{R}^{n\times m}\) and the perturbation \(\eta(t)\) is a \(\mathbb{R}^m\)-valued stationary stochastic process independent of \(\xi(t)\). The author proves that under suitable assumptions the solution of this equation is asymptotically mean square quasi-trivial, i.e., \[ \lim_{j\rightarrow \infty}\mathbf{E}\left[|x(t+j\tau;t_0,\phi)|^2 \right]=0 \] for each \(t\in [t_0,t_0+\tau]\). Furthermore, special cases such as the related linear system and the scalar case are explicitly investigated.