The Regularity of the Linear Drift in Negatively Curved Spaces / François Ledrappier and Lin Shu.

Format:

Book

Author/Creator:

Ledrappier, F., author.

Shu, Lin (Mathematics professor), author.

Series:

Memoirs of the American Mathematical Society ; Volume 281.

Memoirs of the American Mathematical Society Series ; Volume 281

Language:

English

Subjects (All):

Brownian motion processes.

Geodesic flows.

Stochastic analysis.

Physical Description:

1 online resource (164 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is Ck-2 differentiable along any Ck curve in the manifold of Ck Riemannian metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is C1 differentiable along any C3 curve of C3 Riemannian metrics with negative sectional curvature. We formulate the first derivatives of the linear drift and stochastic entropy, respectively, and show they are critical at locally symmetric metrics"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction and statement of results

Main notations and conventions

Chapter 2. Preliminaries

2.1. Jacobi fields and the geodesic flow

2.2. Anosov flow and invariant manifolds

2.3. Harmonic measure for the stable foliation

2.4. Busemann function and the linear drift

Chapter 3. Regularity of the linear drift

3.1. Regularity of the leafwise divergence term ^{ }\overline{ }

3.2. Regularity of the harmonic measure

3.3. Differentials of the linear drift

Chapter 4. Brownian motion and stochastic flows

4.1. Parallelism and the Brownian motion

4.2. A stochastic analogue of the geodesic flow

4.3. Growth of the stochastic tangent maps in time

4.4. Brownian bridge and conditional estimations

4.5. Regularity of the stochastic analogue of the geodesic flow

Chapter 5. The first differential of the heat kernels in metrics

5.1. Strategy

5.2. A description of _{ }^{ }

5.3. The existence of ^{ }_{ }

5.4. Quasi-invariance property of _{ }^{ }

5.5. The extended map ^{ }

5.6. The differential of \mapsto ^{ }( , ,⋅)

Chapter 6. Higher order regularity of the heat kernels in metrics

6.1. A sketch of the proof for Theorem 1.3 with ≥2

6.2. Proofs of the properties concerning ^{ }_{ }

Chapter 7. Regularity of the stochastic entropy

Acknowledgments

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Other Format:

Print version: Ledrappier, François The Regularity of the Linear Drift in Negatively Curved Spaces

ISBN:

9781470473204

1470473208