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!