The Annals of Probability

A Process in a Randomly Fluctuating Environment

Neal Madras

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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.

Article information

Ann. Probab., Volume 14, Number 1 (1986), 119-135.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K99: None of the above, but in this section
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

Randomly fluctuating environment Poisson point process coupling FKG inequality


Madras, Neal. A Process in a Randomly Fluctuating Environment. Ann. Probab. 14 (1986), no. 1, 119--135. doi:10.1214/aop/1176992619.

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