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Title:

An Explicit Local Uniform Large Deviation Bound for Brownian Bridges. submitted

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Abstract. By comparing curve length in a manifold and a standard sphere, we prove a local uniform bound for the exponent in the Large Deviation formula that describes the concentration of Brownian bridges to geodesics. Let M be an m-dimensional complete riemannian manifold with distance function d, Ω(M): = C(R + 0, M) the associated space of con...

Abstract. By comparing curve length in a manifold and a standard sphere, we prove a local uniform bound for the exponent in the Large Deviation formula that describes the concentration of Brownian bridges to geodesics. Let M be an m-dimensional complete riemannian manifold with distance function d, Ω(M): = C(R + 0, M) the associated space of continuous paths and p ∈ M. Let expp denote the exponential map associated to p and by i(p)> 0 its radius of injectivity. Let t> 0 and Q p,q 0,t denote the Brownian Bridge measure on M supported by the set Ω(p, q, t) of paths starting at time 0 in p and ending up at q ∈ M at time t. Provided there is a unique geodesic joining p and q, the tends, as t → 0, to the point measure δγp,q,t supported by this measure Q p,q 0,t geodesic γp,q,t, parametrized proportional to arc-length with constant velocity v(s) = d(p, q)/t. More precisely, we have the following statement. Recall ([3], 3.4, p. 74) that a subset S ⊂ M is called strongly convex, if for any two points q, q ′ ∈ S there is a unique minimizing geodesic joining q and q ′ whose interior is contained in S and ([3], 4.2 Proposition, p.76) that for all p ∈ M there is some number r(p)> 0 such that the geodesic balls B(p, r): = {x ∈ M: d(p, x) < r} are strongly convex for all 0 < r < r(p). Theorem. Let p ∈ M and choose r> 0, ε0> 0 such that R: = r + ε0 < r(p). Let C ⊂ B(p, r) be an arbitrary closed subset. Write κ(C): = max 1, sup q∈C,σ∈G2TqM K(σ) where G2TqM denotes the set of 2-dimensional subspaces in TpM and K(σ) is the sectional curvature of M evaluated at the subspace σ ∈ G2TpM. Then, for all q ∈ C, there is a unique geodesic joining p and q. Furthermore, there is some δ> 0 such that we have for all ε> 0 with ε < ε0 and for all q ∈ C where Q p,q Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://ibb.gsf.de/preprints/2003/pp03-13.pdf

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text

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en

DDC:

515 Analysis *(computed)*

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Title:

An Explicit Local Uniform Large Deviation Bound for Brownian Bridges. submitted

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Abstract. By comparing curve length in a manifold and a standard sphere, we prove a local uniform bound for the exponent in the Large Deviation formula that describes the concentration of Brownian bridges to geodesics. Let M be an m-dimensional complete riemannian manifold with distance function d, Ω(M): = C(R + 0,M) the associated space of cont...

Abstract. By comparing curve length in a manifold and a standard sphere, we prove a local uniform bound for the exponent in the Large Deviation formula that describes the concentration of Brownian bridges to geodesics. Let M be an m-dimensional complete riemannian manifold with distance function d, Ω(M): = C(R + 0,M) the associated space of continuous paths and p ∈ M. Let expp denote the exponential map associated to p and by i(p)> 0 its radius of injectivity. Let t> 0 and Q p,q 0,t denote the Brownian Bridge measure on M supported by the set Ω(p,q,t) of paths starting at time 0 in p and ending up at q ∈ M at time t. Provided there is a unique geodesic joining p and q, the tends, as t → 0, to the point measure δγp,q,t supported by this measure Q p,q 0,t geodesic γp,q,t, parametrized proportional to arc-length with constant velocity v(s) = d(p,q)/t. More precisely, we have the following statement. Recall ([3], 3.4, p. 74) that a subset S ⊂ M is called strongly convex, if for any two points q,q ′ ∈ S there is a unique minimizing geodesic joining q and q ′ whose interior is contained in S and ([3], 4.2 Proposition, p.76) that for all p ∈ M there is some number r(p)> 0 such that the geodesic balls B(p,r): = {x ∈ M: d(p,x) < r} are strongly convex for all 0 < r < r(p). Theorem. Let p ∈ M and choose r> 0, ε0> 0 such that R: = r + ε0 < r(p). Let C ⊂ B(p,r) be an arbitrary closed subset. Write κ(C): = max 1, sup q∈C,σ∈G2TqM K(σ) where G2TqM denotes the set of 2-dimensional subspaces in TpM and K(σ) is the sectional curvature of M evaluated at the subspace σ ∈ G2TpM. Then, for all q ∈ C, there is a unique geodesic joining p and q. Furthermore, there is some δ> 0 such that we have for all ε> 0 with ε < ε0 and for all q ∈ C where Q p,q Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://ibb.gsf.de/preprints/2003/pp03-13.ps

http://ibb.gsf.de/preprints/2003/pp03-13.ps Minimize

Document Type:

text

Language:

en

DDC:

515 Analysis *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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The effective dynamic of a quantum particle on the tubular neighbourhood of some closed submanifold of a riemannian manifold can be described by an effective dynamic on the submanifold as the diameter of the tube decreases to zero. It depends whether Dirichlet- or Neumann

The effective dynamic of a quantum particle on the tubular neighbourhood of some closed submanifold of a riemannian manifold can be described by an effective dynamic on the submanifold as the diameter of the tube decreases to zero. It depends whether Dirichlet- or Neumann Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://ibb.gsf.de/preprints/2003/pp03-07.pdf

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Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

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Construction of surface . . .

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-04-30

Source:

http://www.homepages.ucl.ac.uk/~ucahnsi/Papers/HvWarticle.pdf

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text

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en

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Smooth Homogenization of Heat Equations on Tubular Neighborhoods

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Abstract.We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold of a Riemannian manifold. We show that, as the tube diameter tends to zero, a suitably rescaled and renormalized semigroup converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev index. 1

Abstract.We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold of a Riemannian manifold. We show that, as the tube diameter tends to zero, a suitably rescaled and renormalized semigroup converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev index. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-20

Source:

http://arxiv.org/pdf/0810.5052v1.pdf

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text

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en

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Title:

L 2-Homogenization of Heat Equations on Tubular Neighborhoods

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Abstract.We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold of a Riemannian manifold. We show that, as the tube radius decreases, the semigroup of a suitably rescaled and renormalized generator can be effectively described by a Hamiltonian on the submanifold with a pote...

Abstract.We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold of a Riemannian manifold. We show that, as the tube radius decreases, the semigroup of a suitably rescaled and renormalized generator can be effectively described by a Hamiltonian on the submanifold with a potential that depends on the geometry of the submanifold and of the embedding. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-20

Source:

http://arxiv.org/pdf/0810.5047v1.pdf

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text

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en

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Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-04-30

Source:

http://ibb.gsf.de/preprints/2003/pp03-19.ps

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text

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en

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Brownian Motion close to a Submanifold of a Riemannian manifold

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Abstract.We consider two different ways to force Brownian motion to be close to a submanifold of a riemannian manifold. We investigate their relationship and consider an application to the quantum mechanics of thin layers. 1.

Abstract.We consider two different ways to force Brownian motion to be close to a submanifold of a riemannian manifold. We investigate their relationship and consider an application to the quantum mechanics of thin layers. 1. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://ibb.gsf.de/preprints/2003/pp03-04.ps

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text

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en

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Chernoff’s theorem and discrete time approximations of Brownian motion on manifolds

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-04-30

Source:

http://arxiv.org/pdf/math/0409155v1.pdf

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text

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en

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Title:

The Surface Limit of Brownian Motion in Tubular Neighbourhoods of an embedded Riemannian Manifold

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We construct the surface measure on the space C([0, 1], M) of paths in a compact Riemannian manifold M without boundary embedded into R n which is induced by the usual flat Wiener measure on C([0, 1], R n) conditioned to the event that the Brownian particle does not leave the tubular ε-neighborhood of M up to time 1. We prove that the limit as ε...

We construct the surface measure on the space C([0, 1], M) of paths in a compact Riemannian manifold M without boundary embedded into R n which is induced by the usual flat Wiener measure on C([0, 1], R n) conditioned to the event that the Brownian particle does not leave the tubular ε-neighborhood of M up to time 1. We prove that the limit as ε → 0 exists, the limit measure is equivalent to the Wiener measure on C([0, 1], M), and we compute the corresponding density explicitly in terms of scalar and mean curvature. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://ibb.gsf.de/preprints/2003/pp03-06.pdf

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text

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en

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