We define a de Bruijn process with parameters n and L as a certain continuous-time
Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as
vertices. We determine explicitly its steady state distribution and its characteristic
polynomial, which turns out to decompose into linear factors. In addition, we examine the
stationary state of two specializations in detail. In the first one, the de
Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de
Bruin process, the distribution has constant density but nontrivial correlation functions.
The two point correlation function is determined using generating function techniques.