Asymptotic stability of a system of randomly connected transformations on Polish spaces (Q2724131)

Let \((Y,\rho)\) be a Polish space, \(\Pi_k:{\mathcal R}_+\times Y\to Y\), \(k= 1,\dots, N\), a sequence of continuous transformations, \(\{t_n\}_{n\geq 1}\) a sequence of random variables with independent exponential increments.NEWLINENEWLINENEWLINEThe random dynamical system is defined as follows: \(x_1= \Pi_{k_1}(t_1, x_0)\) where \(k_1\in \{1,\dots, N\}\) is chosen with probability \(p_{k_1}(x_0)\). Next \(x_2= \Pi_{k_2}(t_2- t_1, x_1)\) where \(k_2\) is selected with probability \(p_{k_1k_2}(x_1)\) and so on. Denote NEWLINE\[NEWLINE\mu_n(A)= \text{prob}(x_n\in A),\quad A\in{\mathcal B}(Y),\quad n= 0,1,\dots\;.\tag{1}NEWLINE\]NEWLINE The main result: Assume thatNEWLINENEWLINENEWLINE1) \(\rho(\Pi_k(t, x), \Pi_k(t, y))\leq L_k e^{-\lambda t}\rho(x,y)\), \(L_k>0\), \(\lambda>0\), for \(x,y\in A\), \(t\geq 0\), \(k= 1,\dots, N\), on every bounded set \(A\subset Y\);NEWLINENEWLINENEWLINE2) \(x_*\in Y: \sup\{\rho(\Pi_k(t, x_*): t\geq 0\}< \infty\) for \(k= 1,\dots, N\).NEWLINENEWLINENEWLINEThen there exists a stochastic matrix \(\{p_{ij}(x)\}\) such that \(\mu_n\) defined by (1) is weakly converges to a \(\mu_*\).NEWLINENEWLINENEWLINESo the system under consideration is asymptotically stable.