I was trained in reaction-diffusion (parabolic/elliptic) PDEs, and my research now focuses on applied optimal tranport. I'd like to learn probability and stochastic processes, mostly their connection with PDEs (Feynman-Kac formulas, Itô's calculus, etc.) Could anyone recommend a nice PDE-oriented textbook, anything along the lines of "stochastic methods for PDEs" or whatever? Or at least something that does not require prior knowledge of probability, and giving useful insights for a PDE-oriented reader. I'd like to get the big picture as quickly as possible, so at first I won't care so much about the proofs and sketchy introductory material is fine too.
I'm not too interested in SPDEs per se, more in stochastic representations and techniques for PDEs. So far I came across Richard Bass' textbook "stochastic processes" in the Cambridge Series in Statistical and Probabilistic Mathematics, which I kind of like already, but there might be a better reference out there?
(I know already that someone will want to migrate my thread to stack exchange, please don't?)