## 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

### Keywords:

Random evolution; Markov chain; semigroup of operators
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