Stochastic Analysis on Manifolds

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

“…For a given interval [ [4]) provides an invertible map ϕ u from W(R d ) to W(M). Through ϕ u , Euclidean semi-martingales map to stochastic processes on M. When M is Riemannian and u orthonormal, the result is the Eells-Elworthy-Malliavin construction of Brownian motion [17].…”

“…To handle this, in the diffusion principal component analysis (PCA) framework [1], and with the class of anisotropic normal distributions on manifolds defined in [2,3], data on non-linear manifolds are modelled as being distributed according to transition distributions of anisotropic diffusion processes that are mapped from Euclidean space to the manifold by stochastic development (see [4]). The construction is connected to a non-bracket-generating sub-Riemannian metric on the bundle of linear frames of the manifold, the frame bundle, and the requirement that covariance stays covariantly constant gives a nonholonomically constrained system.…”

Stochastic Taylor expansions and heat kernel asymptotics

“…For further reading on vector fields we refer to the Chapter 1 of [44] and for more explanations on the use of the language of vector fields for stochastic differential equations, we refer to the Chapter 1 of [26].…”

“…We assume here from the reader some basic knowledge on Riemannian manifolds, vector bundles and linear connections. An account on these materials which is designed for probabilists may be found in the book by Hsu [26].…”

“…Bismut in [13] introduces stochastic methods based on Feynman-Kac formula. For probabilistic approaches, we also refer to [30] and Chapter 7 of [26]. For a complete survey on (non probabilistic) heat equation methods for index theorems, we refer to the book [12].…”