General questions on stochastic calculus on manifolds

Question

I've done coursework in differential geometry and stochastic calculus, but I haven't formally seen any connections between the two. I have read that both information geometry and stochastic differential geometry are two different approaches to link the two fields; however, from searching around the internet I haven't settled on any good learning resources that I find accessible.

My questions

1. Any recommendations on a readable book/paper recommendations that specifically link stochastic calculus to manifolds? I'm aware of Emery's book. If there was something that was perhaps slightly more readable, albeit more informal, that would be ideal.

2. Is there a formulation of Feynman-Kac on manifolds?

3. Is there a formulation of the HJB PDE for stochastic control problems on manifolds?

Answer 1

Much of David Elworthy's work is in this general area. For a discussion of the "manifold-valued" Feynman-Kac formula see his text Stochastic Differential Equations on Manifolds (Chapter IX, sec.7) which I find more readable than Emery:

https://www.amazon.com/Stochastic-Differential-Equations-Manifolds-Mathematical/dp/0521287677/

Answer 2

There is also the book by Daniel W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold (American Mathematical Society, 2000). Like Elworthy's book, Stroock strives to rely on general stochastic calculus as little as possible, to the point that the latter's terminology is explicitly restricted to the Introduction.

As far as I remember (it has been about twenty years since I last looked into this text), there were some substantial open problems in the subject by the time this book was published. Some of them are discussed in one of the final sections of the book, ominously (and somewhat amusingly) titled "An admission of defeat". I do not know what is the current status of these problems.