References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. More precisely I'm looking for something that:

1) does a parallelism between the probability of a markov chain, as defined in basic textbooks, with the measure theory counterpart (and the same for other quantities), thus explaining how to go from one to the other

2) shows a possible definition of thermodynamic quantities (such as heat and entropy) in terms of a stochastic process

p.s. this question was taken in part from here

• As a "Markov chains on general state spaces" book, which makes the connection with the countable state space theory, classics are "Markov chains and stochastic stability" by Meyn and Tweedie, and "General Irreducible Markov Chains and Non-Negative Operators" by Nummelin. – Daniel Moskovich Jun 3 '14 at 2:09

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.

Maybe this paper will be helpful: http://www-isl.stanford.edu/~cover/papers/paper103.pdf (Which Processes Satisfy the Second Law?, by T.M. Cover).

A great introduction to probability theory and Markov chains is William Feller's book "An Introduction to Probability Theory and Its Applications" (two volumes).