Is there an overlap between quantum dynamics and math of brownian motion? Suppose you have dynamics of a coherent state. The state presents a normal distribution of finding the particle. Does anyone know of any attempts to connect modern advances in the probability theory to quantum dynamics of a coherent state.
 A: Many different types of connection can be made between stochastic states over commutative algebras of observables and quantum states over noncommutative algebras of observables. As Arnold says, there is a substantial literature.
One approach is to construct both classical and quantum models in a formalism that accommodates both; within the structured environment provided by the joint formalism, one can hope to characterize the difference between classical and quantum. One example of that approach is Lucien Hardy's "Quantum Theory From Five Reasonable Axioms", which, although unpublished in that form and despite its failings, has had a significant impact. I could also self-advertize (again) my approach of constructing classical random fields in a quantum field theoretic formalism (which also, surprise, has failings, but it is rather different from Lucien's and from the Barndorff-Nielsen). Andrei Khrennikov has been trying to construct random field models in a more traditionally stochastic formalism, again with quite different ideas about what he's trying to achieve and making quite different choices of assumptions to modify or to drop.
All the approaches I've touched on above, and that you will find by working through the citations in Andrei Khrennikov's papers, say, are essentially speculative, whereas the Barndorff-Nielsen is more solid as mathematics (his co-author, Richard Gill, is also always interesting when writing alone). What you might find useful depends on your tastes and on what you really intend by your Question.
A: There are tons of papers on the connection between quantum processes and probability theory (though I don't understand why you single out coherent states - they don't play a special role in this connection).
The theory of stochastic processes and the theory of quantum processes are the commutative and noncommutative side of the same coin, with many similarities. 
See, e.g.,  the books by Gardiner (Handbook of stochastic processes) or
Barndorff-Nielsen (Quantum independent increment processes: Structure of quantum Lévy processes, classical probability, and physics)
Online is the following article by Barndorff-Nielsen
http://www.jstor.org/stable/10.2307/3647584
