The asymptotic behaviour of the normalizing factor for random matrix-valued evolution given by a transport equation (Q1605597)

Let \(T^{\varepsilon}(t)\) be a matrix random evolution determined as a solution of the differential equation \[ \frac{dT^{\varepsilon}(t)}{dt}=T^{\varepsilon}(t)A^{\varepsilon}(x(t)) \] with the initial condition \(T^{\varepsilon}(0)=I\), where \(x(t)\) is a regeneration process with moments of regeneration \(\tau_1,\tau_2,\dots,\tau_n,\dots\) and \(\varepsilon\) is a small parameter. \textit{Ya. I. Elejko} and \textit{V. M. Shurenkov} [Ukr. Mat. Zh. 47, No. 10, 1333-1337 (1995; Zbl 0889.60030)] noted that solutions to the equation may be represented in the form \(T^{\varepsilon}(t)=\xi^{\varepsilon}(t)\) for \(0\leq t\leq\tau_1\) and \(T^{\varepsilon}(t)=\xi^{\varepsilon}(\tau_1)\xi^{\varepsilon(1)}(t-\tau_1) \cdots\xi^{\varepsilon(k)}(t-\tau_k)\) for \(\tau_k\leq t\leq\tau_{k+1}\), where \(\xi^{\varepsilon(k)}(t)\) is a solution to the differential equation \[ \frac{d\xi^{\varepsilon(k)}(t)}{dt}= \xi^{\varepsilon(k)}(t)A^{\varepsilon}(x^k(t)) \] with the initial condition \(\xi^{\varepsilon(k)}(0)=I\), and the process \(x^k(t)=x(\tau_k+t),0\leq t<\tau_{k+1}-\tau_k,\) is an independent copy of the process \(x(t),0\leq t\leq\tau\). \textit{Ya. I. Yelejko} and the author [Visn. L'viv. Univ., Ser. Mekh.-Mat. 53, 102-106 (1999; Zbl 0955.60085)] proposed the asymptotic representation of the mean value \(ET^{\varepsilon}(t)\) of the random evolution on the scale of time \(t/\rho^{\varepsilon}\), where \(\rho^{\varepsilon}\to 0\) as \(\varepsilon\to 0\). In this paper the author investigates the asymptotic behaviour of the normalizing factor \(\rho^{\varepsilon}\) that determines the scale of time on which the asymptotic representation for the mean-value \(ET^{\varepsilon}(t)\) of the random evolution was found.