*Europhys. Lett.*,

**65**(5) , pp. 627-632 (2004)

DOI: 10.1209/epl/i2003-10165-4

## Iterated random walk

**L. Turban**

Laboratoire de Physique des Matériaux, UMR CNRS 7556, Université Henri Poincaré (Nancy 1) - BP 239, 54506 Vandoeuvre lès Nancy Cedex, France

(Received 23 September 2003; accepted in final form 15 December 2003)

** Abstract **

The iterated random walk is a random process in which a random
walker moves on a one-dimensional random walk which is itself
taking place on a one-dimensional random walk, and so on. This
process is investigated in the continuum limit using the method
of moments. When the number of iterations
, a
time-independent asymptotic density is obtained. It has a simple
symmetric exponential form which is stable against the
modification of a finite number of iterations. When
*n* is large,
the deviation from the stationary density is exponentially small
in
*n*. The continuum results are compared to Monte Carlo data
for the discrete iterated random walk.

**PACS**

05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.

02.50.-r - Probability theory, stochastic processes, and statistics.

66.30.-h - Diffusion in solids.

**©**

*EDP Sciences 2004*