A Process in a Randomly Fluctuating Environment

Abstract

For every integer $x$, construct a stationary continuous-time Markov process $\gamma(x; t)$, with state space $\{-1, +1\}$ (all processes independent, and having the same distributions). Consider a particle moving at unit speed along the real line, with its direction completely determined by the $\gamma$'s, as follows: if $S_t$ is its position at time $t$, then $S_0 = 0$ and $S_{i + 1} = S_i + \gamma(S_i; i)$ for $i = 0, 1, 2,\cdots$. The increments are not stationary, nor is $S_n$ Markov, yet this process has much in common with the classical random walk, including zero-one laws, a strong law of large numbers, and an invariance principle. The main result of the paper is the proof of the natural conjecture that the process is recurrent if and only if $P\{\gamma(0; 0) = +1\} = \frac{1}{2}$. We also show how the FKG inequality can be used to investigate this process.