Stochastic Calculus in Manifolds

Émery, Michel

doi:10.1007/978-3-642-75051-9

Cited by 387 publications

(443 citation statements)

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“…Several generalizations have already be proposed so far. In the stochastic calculus community, one usually consider the heat kernel p(x, y, t), which is the transition density of the Brownian motion [24,59,60]. This is the smallest positive fundamental solution to the heat equation ∂f ∂t − ∆f =0, where ∆ is the Laplace-Beltrami operator (i.e.…”

Section: An Information-based Generalization Of the Normal Distributionmentioning

confidence: 99%

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and χ 2 law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions.

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Section: An Information-based Generalization Of the Normal Distributionmentioning

confidence: 99%

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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“…Indeed, there exists a family of functions h 1 , ..., h p ⊂ C ∞ (R p ) such that, in the embedded picture, the one form α can be written as α = p j=1 Z j dh j , where Z j ∈ C ∞ (R p ) for j ∈ {1, ..., p}. Therefore, using the properties of the Stratonovich integral (see [E89,Proposition 7.4…”

Section: Proofmentioning

confidence: 99%

“…It can be shown (see [E89,Theorem 6.24]) that there exists a unique linear map θ −→ θ, dX that associates to each such θ a continuous real valued semimartingale and that is fully characterized by the following properties: for any f ∈ C ∞ (M ) and any locally bounded càglàd real-valued process K,…”

Section: Stochastic Integrals Of Forms Along a Semimartingalementioning

confidence: 99%

Stochastic hamiltonian dynamical systems

Lázaro-Camí

Ortega

2008

Reports on Mathematical Physics

145

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We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, are characterized by a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

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“…Fix T > 0. Following [7,Lemma (3.5)] there exists an increasing sequence of predictable stopping times 0…”

Section: Proof Of Theoremmentioning

confidence: 99%

Invariant manifolds for weak solutions to stochastic equations

Filipović¹

2000

Probab Theory Relat Fields

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Abstract. Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C 2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C 2 submanifold, is a strong solution.These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves.

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Stochastic Calculus in Manifolds

Cited by 387 publications

References 0 publications

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Stochastic hamiltonian dynamical systems

Invariant manifolds for weak solutions to stochastic equations

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