## 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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