# Generalized Markov Processes on CW complexes of dimension > 1

Markov processes have a large variety of applications to physics and chemistry (as well as many other fields). Such processes are formulated on graphs, i.e., CW complexes of dimension one. It is possible generalize the notion of Markov process to finite CW complexes $X$ of higher dimension $d$. One way to do this is to employ a "master equation" $$\dot p = H p$$ where $H$ is a suitable time dependent operator on the space of states = the set of $(d-1)$-cells on $X$, the latter a CW complex of dimension $d$. Here $p(t) \in C_{d-1}(X;\Bbb R)$ is a one-parameter family of real cellular $(d-1)$-chains. Here we think of $C_{d-1}(X;\Bbb R)$ as the space of distributions on the set of $(d-1)$-cells (after all, a cellular $(d-1)$-chain amounts to a real-valued function on the the set of $(d-1)$-cells when $X$ is finite.

The operator $H(t): C_{d-1}(X;\Bbb R) \to C_{d-1}(X;\Bbb R)$ is associated with a choice of one-parameter family of functions $E_t: X_{d-1} \to \Bbb R$ and $W_t : X_d\to \Bbb R$, in which $X_k$ is the set of $k$-cells. For reasons of space I will not describe it here (I can refer you for example to the paper: http://front.math.ucdavis.edu/1204.2011 for a description when $d = 1$; it is clear how to generalize the formula to higher dimensions). However, one can think of $H(t)$ as a parametrized family of "biased" diffusion operators, i.e., a family of "combinatorial Laplacians" which is based on modifying the standard inner product structures on $C_{d}(X;\Bbb R)$ and $C_{d-1}(X;\Bbb R)$ using $E$ and $W$ in a suitable way.

Question:

What could this notion possibly have to do with physics or chemistry (or for that matter any of the other scientific disciplines)? Are there any applications of this higher dimensional notion outside of mathematics?

(For example, what about case $d=2$?)

• Is there any particular reason why you insist on your processes being time inhomogeneous?
– R W
Nov 8 '13 at 15:22
• No. Other than the fact that the applications my collaborators and I have in the classical case $d=1$ are in the area of what the physicists term "stochastic pumping." In that scenario external parameters vary periodically in time. Nov 8 '13 at 17:36
• Maybe adding the tag 'pr.probability' to this question would make it more visible to some people who might be able to help. Nov 9 '13 at 19:44
• @Ricardo: good idea. I will. Nov 10 '13 at 0:13
• Maybe I misunderstand the setup, but at least in the case of a finite-dimensional simplicial (or regular CW) complex $X$, wouldn't one be able to reduce everything to a Markov process on the Hasse graph? Nov 15 '13 at 15:23