Here's a simple, exploratory sketch: Imagine an ant which can either move left or right, or stay in its place.
For simplicity, we'll imagine the ant starts at the center of a scale from 1 to 100, and moves one unit at a time.
Ant[k_, u_] :=
If[ u < 1/3, Min[k + 1, 100], If[u > 2/3, Max[0, k - 1], k]]
RecAnt[k_] := Ant[k, RandomReal[]]
{% img center http://farm8.staticflickr.com/7351/122242517234d24311967d.jpg %}
ListPlot[NestList[RecAnt, 50, 100000], ImageSize -> Large, Background -> LightGray]
{% img center http://farm8.staticflickr.com/7418/122244821446ef516fd03z_d.jpg %}