Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries / F. Nier.

Format:

Book

Author/Creator:

Nier, Francis, author.

Series:

Memoirs of the American Mathematical Society ; Volume 252, Number 1200.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 252, Number 1200

Language:

English

Subjects (All):

Manifolds (Mathematics).

Boundary value problems.

Elliptic operators.

Fokker-Planck equation.

Physical Description:

1 online resource (156 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2018]

Summary:

This article is concerned with the maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic laplacian.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Motivations

1.2. The problem

1.3. Main results

1.4. Guidelines for reading this text

Chapter 2. One dimensional model problem

2.1. Presentation

2.2. Results

2.3. Fourier series in ℋ¹ and ℒ²(\rz,| | )

2.4. System of ODE and boundary value problem

2.5. Maximal accretivity

2.6. Extension of the resolvent and adjoint

Chapter 3. Cuspidal semigroups

3.1. Definition and first properties

3.2. Perturbation

3.3. Tensorization

Chapter 4. Separation of variables

4.1. Some notations

4.2. Traces and integration by parts

4.3. Identifying the domains

4.4. Inhomogeneous boundary value problems

Chapter 5. General boundary conditions for half-space problems

5.1. Assumptions for and

5.2. Maximal accretivity

5.3. Half-space and whole space problem

5.4. Resolvent estimates

Chapter 6. Geometric Kramers-Fokker-Planck operator

6.1. Notations and the geometric KFP-operator

6.2. The result by G. Lebeau

6.3. Partitions of unity

6.4. Geometric KFP-operator on cylinders

6.5. Comments

Chapter 7. Geometric KFP-operators on manifolds with boundary

7.1. Review of notations and outline

7.2. Half-cylinders with ∂_{ ¹} ≡0

7.3. Dyadic partition of unity and rescaled estimates

7.4. General local metric on half-cylinders

7.5. Global result

Chapter 8. Variations on a Theorem

8.1. Corollaries

8.2. PT-symmetry

8.3. Adding a potential

8.4. Fiber bundle version

Chapter 9. Applications

9.1. Scalar Kramers-Fokker-Planck equations in a domain of \rz^{ }

9.2. Hypoelliptic Laplacian

Appendix A. Translation invariant model problems

Appendix B. Partitions of unity

Acknowledgements

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4369-4