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A process in a randomly fluctuating environment. (English) Zbl 0593.60099

Let \(\{\) \(\gamma\) (x,t), t\(>0\}\) be stationary Markov processes with parameter x (independent, having the same distribution) on the state space \(\{-1,+1\}\) with transition probability matrix \[ \left[ \begin{matrix} P_{-1,-1}(t),\quad P_{-1,+1}(t)\\ P_{+1,-1}(t),\quad P_{+1,+1}(t)\end{matrix} \right]=\left[ \begin{matrix} q+pr^ t,\quad p-pr^ t\\ q-qr^ t,\quad p+qr^ t\end{matrix} \right] \] where \(p=\beta /(\beta +\delta)\), \(q=\delta /(\beta +\delta)\), and \(r=\exp (-\beta -\delta)\). Define \(S_ 0=0\); \(X_{i+1}=\gamma (S_ i,i),\quad i=0,1,...;\quad S_ t=S_ i+(t-i)X_{i+1},\) \(i<t\leq i+1\). It is shown that although \(\{S_ n\}\) is not Markovian, the process which represents a moving particle in a randomly fluctuating environment has many of the properties of classical random walks, such as zero-one laws, strong law of large numbers and invariance principle.
The main theorem of this paper is that: \(\{S_ n\}\) is recurrent if and only if \(\beta =\delta\). Although this is a natural conjecture if we consider the case of symmetric random walk, its proof is not very simple and uses coupling and the one-dimensional nearest-neighbor techniques. As the author remarks, the process can be generalized to \(d>1\) dimension, however, the criterion for recurrence of the general model \(\{S_ n\}\) has not been discussed.
Reviewer: Chengxun Wu

MSC:

60K99 Special processes
60J27 Continuous-time Markov processes on discrete state spaces
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G50 Sums of independent random variables; random walks
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