Crate cartan_stochastic 

cartan-stochastic

Stochastic analysis primitives on Riemannian manifolds.

This crate provides the foundation that downstream crates (hsu, bismut, elworthy, malliavin) need to do probability, SDE integration, and pathwise-derivative computation on manifolds — independent of the underlying manifold type, as long as it implements the cartan-core Manifold + ParallelTransport + Retraction trait stack.

The architectural purpose is to prevent primitive duplication across the Hsu / Bismut / Elworthy / Malliavin stack. Horizontal lift, orthonormal frame bundle, Stratonovich development, and stochastic-development by Euler-Maruyama are all defined once here.

Concepts

Orthonormal frame bundle O(M): the total space of orthonormal bases of the tangent spaces of M. A point in O(M) is a pair (p, r) where p ∈ M and r = (e_1, …, e_n) is an orthonormal basis of T_p M.

Horizontal lift: given a tangent vector u ∈ T_p M, a curve in O(M) whose velocity projects to u and whose frame evolves by parallel transport. Implemented as a right action of R^n on the frame bundle via (p, r) · ξ = (γ(1), r̃) where γ is the exponential of Σ ξ_i e_i and r̃ is the parallel transport of r along γ.

Stochastic development (Eells-Elworthy-Malliavin): solve the SDE on O(M) driven by Euclidean Brownian motion W_t, with Stratonovich differential ∂_t (p, r) = H_i(p, r) ∘ dW^i_t where H_i is the horizontal lift of the i-th frame vector. The projection to M is Brownian motion on M with the Laplace-Beltrami generator.

Minimum trait requirements

Any manifold implementing Manifold + ParallelTransport + Retraction can host a stochastic development. Exact exponentials are not required; retraction suffices at the cost of higher-order discretisation error.

References

• Hsu, Elton P. Stochastic Analysis on Manifolds. AMS, 2002. Chapter 2 (horizontal lift and anti-development), Chapter 3 (Brownian motion via orthonormal frame bundle).
• Eells, J. and Elworthy, K. D. Wiener integration on certain manifolds. Problems in Non-Linear Analysis, 1971.
• Elworthy, K. D. Stochastic Differential Equations on Manifolds. Cambridge LMS Lecture Notes 70, 1982.

Re-exports

pub use development::stochastic_development;

pub use development::DevelopmentPath;

pub use error::StochasticError;

pub use frame::random_frame_at;

pub use frame::OrthonormalFrame;

pub use horizontal::horizontal_velocity;

pub use sde::stratonovich_step;

pub use sde::StratonovichDevelopment;

pub use wishart::wishart_step;

Modules

development

Stochastic development (Eells-Elworthy-Malliavin).

error

Error types for stochastic-analysis primitives.

frame

Orthonormal frames on Riemannian manifolds.

horizontal

Horizontal lift from R^n to the tangent bundle via an orthonormal frame.

sde

Stratonovich stochastic differential equations on Riemannian manifolds.

wishart

Wishart Brownian motion on the SPD cone.