A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold

Driver, Bruce K.

doi:10.1016/0022-1236(92)90035-h

Cited by 208 publications

(193 citation statements)

References 16 publications

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“…Related results in this context and a detailed discussion of the Dirichlet forms which arise can be found in Elworthy-Ma [26] and [22]. See also Driver [9] and Hsu [29]. …”

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

confidence: 92%

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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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“…Related results in this context and a detailed discussion of the Dirichlet forms which arise can be found in Elworthy-Ma [26] and [22]. See also Driver [9] and Hsu [29]. …”

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

confidence: 92%

“…Note that if E = T M condition (M) holds with ·, · ′ = ·, · if and only if ∇ is torsion skew symmetric as described by Driver in [9]. In particular Condition (M ) holds for the SDE's in Examples 1-3, section 2.1.1.…”

Section: The Covariant Differentiation Operator Id Dtmentioning

confidence: 93%

Itô maps and analysis on path spaces

Elworthy

2007

Math. Z.

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“…Moreover, the applications of these ideas is not restricted to the generation of estimates on p T ( · , y). For instance, throughout the last decade, Malliavin [FG] and his school have been applying them to the analysis of diffusions on loop spaces, and, more recently, B. Driver ([Dr1] and [Dr2]) has successfully constructed the number operator on loop space. (See also [H], [ES1], and [ES2].…”

Section: Y)mentioning

confidence: 99%

Untitled

1996

Bull. Amer. Math. Soc.

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Abstract. This article is intended to provide non-specialists with an introduction to integration theory on pathspace. §0: Introduction Let Q 1 denote the set of rational numbers q ∈ [0, 1], and, for a ∈ Q 1 , define the translation map τ a on Q 1 by addition modulo 1:Is there a probability measure λ Q1 on Q 1 which is translation invariant in the sense thatTo answer this question, it is necessary to be precise about what it is that one is seeking. Indeed, the answer will depend on the level of ambition at which the question is asked. For example, if A denotes the algebra of subsets Γ ⊆ Q 1 generated by intervalsfor a ≤ b from Q 1 , then it is easy to check that there is a unique, finitely additive way to define λ Q1 on A. In fact,On the other hand, if one is more ambitious and asks for a countably additive λ Q1 on the σ-algebra generated by A (equivalently, all subsets of Q 1 ), then one is asking for too much. In fact, since λ Q1 ({q}) = 0 for every q ∈ Q 1 , countable additivity forceswhich is obviously irreconcilable with λ Q1 (Q 1 ) = 1. Thus, in order to achieve countable additivity, one has to leave Q 1 behind. In particular, one can complete Q 1 and pass to [0, 1], where Lebesgue and his measure come to the rescue. Of course, one pays a heavy price for this transition: Q 1 is invisible to Lebesgue measure!

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“…There is also the group of extensions of the Cameron-Martin theorem used in the study of loop groups; see Albeverio and Hoegh-Krohn [2], Frenkel [16], Gross [22], and Malliavin and Malliavin [35]. In Driver [13] it is shown that the classical Cameron-Martin theorem extends to the case of compact Riemannian manifolds (see Theorem 3.1), which includes Wiener measure on the path space W(G) of a compact Lie group G. The purpose of this paper is to extend the results in [ 13] to the case of "pinned Wiener measure" on a compact Riemannian manifold M ; see Proposition 3.3 and Theorem 3.4. We also derive an integration by parts formula for "A-derivatives"; see Theorem 3.13.…”

Section: Introductionmentioning

confidence: 99%

A Cameron-Martin Type Quasi-Invariance Theorem for Pinned Brownian Motion on a Compact Riemannian Manifold

Driver¹

1994

Transactions of the American Mathematical Society

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A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold

Cited by 208 publications

References 16 publications

Itô maps and analysis on path spaces

Itô maps and analysis on path spaces

Untitled

A Cameron-Martin Type Quasi-Invariance Theorem for Pinned Brownian Motion on a Compact Riemannian Manifold

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