PDE-oriented textbook on probability and random processes?

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?

• At the risk of going off on a tangent, you might enjoy browsing through Chirikjian's two volumes: "Stochastic Models, Information Theory and Lie Groups". – J W Feb 17 '18 at 8:04

Stochastic processes and application by Pavliotis is a good one.

• thank you Piyush Grover, this is exactly the kind of stuff I was looking for. Great reference (at least after a quick glance)! – leo monsaingeon Feb 17 '18 at 17:15

Maybe, you can have a look at this book:

Second order PDE’s in finite and infinite dimensions. A probabilistic approach, S. Cerrai

In many classical text books in probability, there are one or two chapters regarding the link between PDE and Brownian motion.

Finally, there is this book which might be useful to you

An introduction to partial differential equation for probabilists, D.W. Stroock

• Thanks. I suspect indeed that "there are one or two chapters regarding the link between PDE and Brownian motion" in any probability textbook, but that's precisely NOT what I'm looking for: I don't want to have to learn a whole lot of probability for the sake of it before I can start digging into connections with PDEs for a couple of chapters only. I'll look into Stroock's book ASAP, though. – leo monsaingeon Feb 17 '18 at 17:17

Pao-Liu Chow - Stochastic Partial Differential Equations, 2007

In there you could find topics such as Ito's formula, Feynman-Kac formula and also stochastic parabolic equations. And much more. This is one of the first books I've read when I started to learn more about connections of PDEs and stochastic processes. It helped me a lot.

• Does it pertains PDE related probabilistic techniques, or only SPDE topics? The OP is not interested in SPDE... – Amir Sagiv Jun 2 '18 at 16:40
• It has both :) Also I saw it yesterday that there is a new 2014 edition that has more applications included. – Mark Jun 4 '18 at 8:40