On the Geometry of Diffusion Operators and Stochastic Flows

Elworthy, K. D.; Jan, Y Le; Li, Xue Mei

doi:10.1007/bfb0103064

Cited by 99 publications

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“…In particular Condition (M ) holds for the SDE's in Examples 1-3, section 2.1.1. For examples where it does not hold see Elworthy-LeJan-Li [15], in which there is also the following result (Proposition 3.3.11, p72): From this we see immediately that condition (M ) implies condition (M 0 ) in the compact case under consideration.…”

Section: The Covariant Differentiation Operator Id Dtmentioning

confidence: 72%

“…The result on which, Elworthy-LeJan-Li [15], and this article are based is the following, c.f. Quillen [40], Narasimhan-Ramanan [37]: [17], [15], c.f. Quillen [40]) For each such map X : R m × M → T M there is a unique connection∇ on E such that…”

Section: An Sde Which Induces the Connection ∇mentioning

confidence: 99%

“…as described in Elworthy-Yor [23] for gradient systems and Elworthy-LeJan-Li [17], [15] more generally. Here…”

Section: The Derivative T I Of the Itô Map And T Imentioning

confidence: 99%

“…From Elworthy-LeJan-Li [15] and Elworthy-Ma [26] and the compactness of M , for almost all σ the inclusion H σ ֒→ T σ C x0 M is continuous and its norm is in L p as a function of σ for all p ∈ [1, ∞) and is essentially bounded for any of these Finsler norms if condition (M) holds. A function f : C x0 M → R will be said to be BC 1 if it is Fréchet differentiable and is bounded together with its differential df considered as a section of T * C x0 M , again using any of these Finsler structures.…”

Section: Sobolev Calculus On C X 0 M and Its Intertwining By Itô Mapsmentioning

confidence: 99%

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Itô maps and analysis on path spaces

Elworthy

2007

Math. Z.

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We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, I, of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given I, is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.

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Section: The Covariant Differentiation Operator Id Dtmentioning

confidence: 72%

Section: An Sde Which Induces the Connection ∇mentioning

confidence: 99%

“…as described in Elworthy-Yor [23] for gradient systems and Elworthy-LeJan-Li [17], [15] more generally. Here…”

Section: The Derivative T I Of the Itô Map And T Imentioning

confidence: 99%

Section: Sobolev Calculus On C X 0 M and Its Intertwining By Itô Mapsmentioning

confidence: 99%

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Itô maps and analysis on path spaces

Elworthy

2007

Math. Z.

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“…The sample space (U ε , γ ε ) as defined above has a simple probabilistic interpretation. A random section u ε ∈ U ε is a random linear superposition 16) where the coefficients X ε n are independent normal random variables with mean 0 and variances…”

Section: Theorem 14mentioning

confidence: 99%

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

Nicolaescu

Savale

2016

Trans. Amer. Math. Soc.

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Abstract. We prove that the Euler form of a metric connection on a real oriented vector bundle E over a compact oriented manifold M can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on E from the statistics of random sections of E.

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A Class of Vector Fields on Path Spaces

Lyons

Qian

1997

Journal of Functional Analysis

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In this paper we show that the vector field X {, h on a based path space W o (M) over a Riemannian manifold M defined by parallel translating a curve h in the initial tangent space T o M via an affine connection { induces a solution flow which preserves the Wiener measure on the based path space W o (M), provided the affine connection { is adjoint skew-symmetric. In the case when { is a metric connection, then { is adjoint skew-symmetric if and only if { is torsion skew-symmetric.1997 Academic Press

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On the Geometry of Diffusion Operators and Stochastic Flows

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 99 publications

References 0 publications

Itô maps and analysis on path spaces

Itô maps and analysis on path spaces

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

A Class of Vector Fields on Path Spaces

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