Reference Request: Probability and (Nonlinear) PDEs I'm a graduate student interested in learning about probability and (mostly evolutionary) PDEs, just for fun (and as an excuse to learn some probability).  I'm mostly interested in things along the lines of: what can we (almost surely) prove for randomized (if that's the right word) initial data which fails (or has yet to be proven) in the deterministic setting.  I'm less interested in, e.g., stochastic PDE.  I'm also not entirely opposed to there being some elliptic theory, but I'd like the focus to be primarily on evolution equations.
Ideally I'd like to read a book (or (series of) paper(s)) that covers a variety of topics and gives (with proof) "representative" results without getting too bogged down in details (e.g., in the name of maximal generality); something like Tao's Nonlinear Dispersive Equations (but for the probabilistic setting).  I've learned standard linear and nonlinear (deterministic) PDE theory and (consequently) real analysis/measure theory, but no probability.  The PDE book wouldn't have to cover probability, but if it doesn't, I'd appreciate a suggested probability book to learn what I'd need to understand the PDE material.  Thanks!
 A: A recent book more or less of the kind you want is
V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, Berlin, 2011.
Probabilistic methods for nonlinear partial differential equations are developed in a series of papers by Yana Belopolskaya, for example
Y. Belopolskaya and W. Woyczynski, Generalized solutions of nonlinear parabolic equations and diffusion processes. Acta Appl. Math. 96, No. 1-3, 55-69 (2007).
A: Caveat: I am not an expert on the subject. My knowledge of it consists of having read a paper or two, private conversations with three or four actual experts a few years ago, and sitting through five or six lectures/seminars on the topic. So the summary below is very biased and incomplete. 
Caveat 2: I hope I am interpreting correctly what you meant by "randomised initial data".

I am not aware of any books that cover what you are interested in. The field is still quite active and I am not aware of anyone actually sitting down and summarising the development in the past 20 years or so. 
A somewhat recent review article that is sufficiently introductory was written by Burq and Tzvetkov. You should also consult its references. 
A paper that everyone refers to is Lebowitz, Rose, and Speer; it will give you at least the initial point of view of studying invariant Gibbs measures for nonlinear dispersive equations. In terms of the pure mathematics, much of the early breakthroughs seem to be due to Bourgain 1 2 3. He gave a quick review of the state of the art in 2000. 
Since then, there has been much more development for construction of the invariant Gibbs measure and almost sure existence of solution below critical regularity. Some of the names of people whom I know worked on, or are working on, the subject include (in no particular order): Jean Bourgain, Tadahiro Oh, Nicolay Tzvetkov, Aynur Bulut, Nicolas Burq, Kay Kirkpatrick... Anyone of those people would be more qualified than I to answer this question (unfortunately none of them appears to be on MathOverflow). You should check the arXiv for their papers on the subject, and I am pretty sure Tadahiro and Aynur at least (whom I know better personally) will be happy to answer some of your questions by e-mail. 
Last but not least, you may want to check out Staffilani's slides on the subject: 1 2. (The second link I just stumbled upon now while looking for the first one; it looks like a nice introduction!)
