  We use algebraic topology to study the stochastic mo- tion of cellular cycles
in a finite CW complex. Inspired by statis- tical mechanics, we introduce a
homological observable called the average current. The latter measures the
average flux of the prob- ability in the process. In the low temperature,
adiabatic limit, we prove that the average current fractionally quantizes, in
which the denominators are combinatorial invariants of the CW complex.
