Stochastic Analysis on Manifolds
The cartan-stochastic crate extends the cartan stack from deterministic geometry to stochastic dynamics on Riemannian manifolds. It provides the primitives that Brownian-motion, SDE, and pathwise-derivative machinery all depend on — implemented once and shared across every downstream crate that needs them.
Layer map
The stochastic layer (L1) sits between the geometry primitives (L0) and any probabilistic or financial application built on top:
| Layer | Crate | Responsibility |
|---|---|---|
| L0 | cartan-core, cartan-manifolds, cartan-geo, cartan-optim, cartan-dec | Manifolds, connections, curvature, Jacobi fields, retraction, parallel transport, DEC. |
| L1 | cartan-stochastic | Orthonormal frame bundle, horizontal lift, Stratonovich SDE, stochastic development, Wishart process. |
| L2 | hsu (external) | Heat-kernel estimates, Feynman-Kac, path-space integration. |
| L3 | bismut, elworthy (external) | Bismut formula, BEL weights, Malliavin matrix. |
| L4 | malliavin (external) | Greeks, pricing, strategy sizing. |
What L1 provides
Design
L1 depends on cartan-core only through Manifold + Retraction + VectorTransport. Any manifold satisfying that bound — including manifolds outside the cartan-manifolds crate — automatically inherits Brownian-motion integration via the generic stochastic_development routine. The sphere, affine-invariant SPD, and Bures-Wasserstein SPD are all tested end-to-end out of the box.
No ambient Euclidean embedding, no explicit chart atlas, no hand-written Itô drift is required at the call site: the Eells-Elworthy-Malliavin construction (rolling without slipping on the orthonormal frame bundle) produces the correct Laplace-Beltrami generator from pure Stratonovich frame dynamics.
References
- Hsu, Elton P. Stochastic Analysis on Manifolds. AMS, 2002.
- Elworthy, K. D. Stochastic Differential Equations on Manifolds. Cambridge LMS Lecture Notes 70, 1982.
- Eells, J. and Elworthy, K. D. Wiener integration on certain manifolds. Problems in Non-Linear Analysis, 1971.