I have had this question in my head for a long time, and if I don't find out the answer I may explode.
So I'm familiar with a basic Ito process, let's say:
$dX_t = \mu d t + \sigma d Z_t$.
There is plenty of theory about this drift-diffusion. In fact, given the parameters, I know the probability distribution of $X_T$ for any $T$. Furthermore, I can use Ito's lemma to determine the probability distribution of a function of $X_T$ -- even more valuable.
Here's what I'm interested in: say I want to define a stochastic process of a random variable taking values in a group. For example, I might want to define some stochastic process on $\mathbb{Z}/n\mathbb{Z}$, or maybe on $D_{12}$, or whatever. But the group need not be finite -- maybe I want to define my stochastic process over $GL_n(\mathbb{R})$ somehow.
I've given considerable thought to how one could formulate these mechanics but the problem rapidly escalates beyond my understanding of either algebra or probability theory. I know I could use Poisson process simulations to obtain results, but I feel there must be someone who asked this question first and laid some theoretical groundwork I could use in my studies.
I've investigated random walks on Cayley groups, free probability and some other stuff, but in most cases understood little except that the results were either very general or almost trivial. I'm looking for something far more powerful -- it'd need a way to assign probabilities to group elements, a way to describe the probability of a group element "happening" in continuous time, etc.
So my questions are really:
Is there any pre-existing theory/understanding of stochastic processes over group-valued random variables, and functions on those group-valued random variables? And
If there isn't, which parts of group theory or probability theory could I use to begin formulating something like this?
Or, is the notion of a group simply too general to derive any meaningful result?
I'd be interested in answers to any of these questions, or any thoughts/comments on the idea of group-valued continuous-time stochastic processes.