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I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.

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If you're looking for an approach focusing on applications in engineering, there are Chirikjian's two volumes:

Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods

Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications

Chirikjian doesn't presuppose much mathematical background beyond linear algebra, multivariable calculus and differential equations, but I would tend to recommend also having studied at least basic probability for a smoother start.

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Check out "Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing" by Pierre Henry-Labord$\grave{\text{e}}$re. While this book is very focused on applications, it has an interesting blend of both advanced ideas in manifold theory (connections on vector bundles) as well probability and stochastic processes. The only unfortunate part is that I can't say this book is all that axiomatic, but if you're somewhat comfortable with both subjects that this might give you one avenue to start integrating ideas from both.

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Personally, I found Hsu's A Brief Introduction to Brownian Motion on a Riemannian Manifold to be a really nice introduction, which can be continued via the same author's full book on the subject, Stochastic Analysis on Manifolds.

There is also Watanabe and Ikeda's Stochastic Differential Equations and Diffusion Processes, which has a large and useful (for me) section on stochastic Ito diffusion processes on manifolds, but also has a great deal of introductory material leading up to it (on the stochastic side at least).

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