I asked myself the same question (starting from a different background) a while ago. Here is where I stand.
1) A standard reference on functional analytic treatments of stochastic processes is the book by Kallenberg. Another huge and free reference is the series of book by Fremlin. Stochastic processes can be specified in a variety of inequivalent ways, depending on your preference of background. There are:
- the master equation approach, reminiscent of the Schrodinger equation in QM
- the Markov kernel approach, where a Markov chain is given as a transition probability from a point to a measurable set
- path integrals, stochastic integrals and Langevin equations, i.e. measure theoretic treatments of spaces of trajectories (see the book by Oksendal).
There are of course bridges between each approaches, e.g. the Feynman-Kac formula for diffusion processes. It is a huge and fascinating subject. For a mathematically precise presentation of statistical dynamics, I could also recommend the book by Streater, "Statistical Dynamics".
2) You should maybe take a look at the work on the "Fluctuation-Dissipation" relations, a possible starting point is this article by Jarzynski and the work of Ken Sekimoto. I find this page a helpful reference: Extremal principles in non-equilibrium thermodynamics . For standard (equilibrium) thermodynamics, the book of Streater is not bad either.