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Random evolutions processes induced by discrete time Markov chains. (English) Zbl 0946.47029

Let \(X(n)\) be a stationary Markov chain with \( N-\) states and \({T_i,1 \leq i \leq N}\) be bounded operators on a Banach space \(B\). Define the backward random evolution \( R(0)=I, R(n)=T_{X(0)} \cdots T_{X(n)}\), \(1 \leq n \) and \(\widetilde{R(n)}\) by \((\widetilde{R(n)} \widetilde{f})_j=E_J [R(n)f_{X(n)}]\), \(\widetilde{f} \in B^N\). Then \((\widetilde{R(n)} \widetilde{f})\) is a solution of the following equations \[ u_j(n+1)= \sum_{k=1}^Np_{jk}T_ju_k(n). \] A corresponding forward version defined by \( S(0)=I, S(n)=T_{X(n)} \cdots T_{X(0)}\), \(1 \leq n\) also enjoys similar propriety. And the reversed chain \(\widehat{X}(n)\) defined by the invariant measure also induces backward and forward random evolutions \(\widehat{R}(n)\) and \(\widehat{S}(n)\). Similar results hold too. It is then extended to a jump backward random evolution defined by \[ R(0)=I, R(n)=T_{X(0)}C_{X(0)X(1)}T_{X(1)} \cdots C_{X(n-1)X(n)}T_{X(n)}, 1 \leq n , \] and its forward version, where \({C_{kl}}, 1 \leq k\), \(l \leq N \), are bounded linear operators on the Banach space \( B \). At the end of this paper, the result is extended to the case where the Markov chain \(X(n)\) is nonstationary.

MSC:

47D07 Markov semigroups and applications to diffusion processes
60J25 Continuous-time Markov processes on general state spaces
60F05 Central limit and other weak theorems
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