cartan-stochastic

Stochastic analysis primitives on Riemannian manifolds.

Part of the cartan workspace.

What this crate does

cartan-stochastic provides the foundation downstream stochastic stacks need to do probability, SDE integration, and pathwise-derivative computation on manifolds — independent of the manifold type, as long as it implements cartan-core's Manifold + ParallelTransport + Retraction.

The architectural purpose is to prevent primitive duplication across the Hsu / Bismut / Elworthy / Malliavin stack: horizontal lift, the orthonormal frame bundle O(M), Stratonovich development, and Euler- Maruyama stochastic development are defined once here.

Core constructs

Orthonormal frame bundle O(M) — the total space of orthonormal bases of the tangent spaces of M. A point is (p, r) where r = (e_1, …, e_n) is an orthonormal basis of T_p M.
Horizontal lift — given 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.
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. Pushed-down trajectory on M is Brownian motion in the Riemannian sense.
Wishart SPD diffusion — closed-form SDE on the SPD manifold, used by cartan-homog's stochastic ensembles for Wishart-perturbed phase property propagation.

Downstream consumers

cartan-homog's WishartRveEnsemble (stochastic feature): perturb one phase's property along a Wishart trajectory, aggregate the effective tensors with the Karcher (Frechet) mean on Spd<N>.
Reserved foundation for future Bismut-Elworthy-Li Greeks work.

Example

use cartan_manifolds::Spd;
use cartan_stochastic::WishartSpdDiffusion;
use rand::SeedableRng;

let manifold = Spd::<3>::new();
let mut rng = rand::rngs::StdRng::seed_from_u64(0);

let p0 = manifold.identity();
let diff = WishartSpdDiffusion::new(/* parameters */);
// step the SDE forward in time

License