Analysis for Diffusion Processes on Riemannian Manifolds.

By: Wang, Feng-YuMaterial type: TextTextSeries: Advanced Series on Statistical Science and Applied Probability SerPublisher: Singapore : World Scientific Publishing Co Pte Ltd, 2014Copyright date: ©2013Description: 1 online resource (392 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789814452656Subject(s): Diffusion processes | Riemannian manifoldsGenre/Form: Electronic books.Additional physical formats: Print version:: Analysis for Diffusion Processes on Riemannian ManifoldsDDC classification: 516.373 LOC classification: QA649.W364 2014ebOnline resources: Click to View
Contents:
Intro -- Contents -- Preface -- 1. Preliminaries -- 1.1 Riemannian manifold -- 1.1.1 Differentiable manifold -- 1.1.2 Riemannian manifold -- 1.1.3 Some formulae and comparison results -- 1.2 Riemannian manifold with boundary -- 1.3 Coupling and applications -- 1.3.1 Transport problem and Wasserstein distance -- 1.3.2 Optimal coupling and optimal map -- 1.3.3 Coupling for stochastic processes -- 1.3.4 Coupling by change of measure -- 1.4 Harnack inequalities and applications -- 1.4.1 Harnack inequality -- 1.4.2 Shift Harnack inequality -- 1.5 Harnack inequality and derivative estimate -- 1.5.1 Harnack inequality and entropy-gradient estimate -- 1.5.2 Harnack inequality and L2-gradient estimate -- 1.5.3 Harnack inequalities and gradient-gradient estimates -- 1.6 Functional inequalities and applications -- 1.6.1 Poincar e type inequality and essential spectrum -- 1.6.2 Exponential decay in the tail norm -- 1.6.3 The F-Sobolev inequality -- 1.6.4 Weak Poincare inequality -- 1.6.5 Equivalence of irreducibility and weak Poincare inequality -- 2. Diffusion Processes on Riemannian Manifolds without Boundary -- 2.1 Brownian motion with drift -- 2.2 Formulae for Pt and RicZ -- 2.3 Equivalent semigroup inequalities for curvature lower bound -- 2.4 Applications of equivalent semigroup inequalities -- 2.5 Transportation-cost inequality -- 2.5.1 From super Poincare to weighted log-Sobolev inequalities -- 2.5.2 From log-Sobolev to transportation-cost inequalities -- 2.5.3 From super Poincare to transportation-cost inequalities -- 2.5.4 Super Poincare inequality by perturbations -- 2.6 Log-Sobolev inequality: Different roles of Ric and Hess -- 2.6.1 Exponential estimate and concentration of -- 2.6.2 Harnack inequality and the log-Sobolev inequality -- 2.6.3 Hypercontractivity and ultracontractivity -- 2.7 Curvature-dimension condition and applications.
2.7.1 Gradient and Harnack inequalities -- 2.7.2 HWI inequalities -- 2.8 Intrinsic ultracontractivity on non-compact manifolds -- 2.8.1 The intrinsic super Poincare inequality -- 2.8.2 Curvature conditions for intrinsic ultracontractivity -- 2.8.3 Some examples -- 3. Reflecting Diffusion Processes on Manifolds with Boundary -- 3.1 Kolmogorov equations and the Neumann problem -- 3.2 Formulae for Pt, RicZ and I -- 3.2.1 Formula for Pt -- 3.2.2 Formulae for RicZ and I -- 3.2.3 Gradient estimates -- 3.3 Equivalent semigroup inequalities for curvature conditionand lower bound of I -- 3.3.1 Equivalent statements for lower bounds of RicZ and I -- 3.3.2 Equivalent inequalities for curvature-dimension condition and lower bound of I -- 3.4 Harnack inequalities for SDEs on Rd and extension to nonconvex manifolds -- 3.4.1 Construction of the coupling -- 3.4.2 Harnack inequality on Rd -- 3.4.3 Extension to manifolds with convex boundary -- 3.4.4 Neumann semigroup on non-convex manifolds -- 3.5 Functional inequalities -- 3.5.1 Estimates for inequality constants on compact manifolds -- 3.5.2 A counterexample for Bakry-Emery criterion -- 3.5.3 Log-Sobolev inequality on locally concave manifolds -- 3.5.4 Log-Sobolev inequality on non-convex manifolds -- 3.6 Modified curvature tensors and applications -- 3.6.1 Equivalent semigroup inequalities for the modified curvature lower bound -- 3.6.2 Applications of Theorem 3.6.1 -- 3.7 Generalized maximum principle and Li-Yau's Harnack inequality -- 3.7.1 A generalized maximum principle -- 3.7.2 Li-Yau type gradient estimate and Harnack inequality -- 3.8 Robin semigroup and applications -- 3.8.1 Characterization of PtQW and D(EQ) -- 3.8.2 Some criteria on Q for (M) = 1 -- 3.8.3 Application to HWI inequality -- 4. Stochastic Analysis on Path Space over Manifolds with Boundary -- 4.1 Multiplicative functional.
4.2 Damped gradient, quasi-invariant ows and integration by parts -- 4.2.1 Damped gradient operator and quasi-invariant flows -- 4.2.2 Integration by parts formula -- 4.3 The log-Sobolev inequality -- 4.3.1 Log-Sobolev inequality on WTx -- 4.3.2 Log-Sobolev inequality on the free path space -- 4.4 Transportation-cost inequalities on path spaces over convex manifolds -- 4.5 Transportation-cost inequality on the path space over nonconvex manifolds -- 4.5.1 The case with a diffusion coefficient -- 4.5.2 Non-convex manifolds -- 5. Subelliptic Diffusion Processes -- 5.1 Functional inequalities -- 5.1.1 Super and weak Poincare inequalities -- 5.1.2 Nash and log-Sobolev inequalities -- 5.1.2.1 Heat kernel estimate -- 5.1.2.2 Nash and log-Sobolev inequalities -- 5.1.3 Gruschin type operator -- 5.1.3.1 Weak Poincare inequality -- 5.1.3.2 Super Poincare inequality -- 5.1.3.3 Nash inequality -- 5.1.3.4 Log-Sobolev inequality -- 5.1.4 Kohn-Laplacian type operator -- 5.1.4.1 Weak Poincare inequality -- 5.1.4.2 Super Poincare inequality -- 5.1.4.3 Nash inequality -- 5.1.4.4 Log-Sobolev inequality -- 5.2 Generalized curvature and applications -- 5.2.1 Derivative inequalities -- 5.2.2 Applications of Theorem 5.2.1 -- 5.2.2.1 L2-derivative estimate and applications -- 5.2.2.2 Entropy-derivative estimate and applications -- 5.2.2.3 Exponential decay and Poincare inequality -- 5.2.2.4 Derivative inequalities by (5.2.3) -- 5.2.3 Examples -- 5.2.4 An extension of Theorem 5.2.1 -- 5.3 Stochastic Hamiltonian system: Coupling method -- 5.3.1 Derivative formulae -- 5.3.2 Gradient estimates -- 5.3.2.1 | Z| is bounded -- 5.3.2.2 | Z| has polynomial growth -- 5.3.2.3 A general case -- 5.3.3 Harnack inequality and applications -- 5.3.3.1 Harnack inequality under (A5.3.1) -- 5.3.3.2 Harnack inequality under assumption (A5.3.2).
5.3.4 Integration by parts formula and shift Harnack inequality -- 5.4 Stochastic Hamiltonian system: Malliavin calculus -- 5.4.1 A general result -- 5.4.2 Explicit formula -- 5.4.3 Two specific cases -- 5.4.3.1 Rank[B0] = m -- 5.4.3.2 A := (1)Z(1) is constant -- 5.5 Gruschin type semigroups -- 5.5.1 Derivative formula -- 5.5.2 Log-Harnack inequality -- 5.5.2.1 Construction of Yt(1) -- 5.5.2.2 Construction of Yt(2) -- Bibliography -- Index.
Summary: Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary.
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Intro -- Contents -- Preface -- 1. Preliminaries -- 1.1 Riemannian manifold -- 1.1.1 Differentiable manifold -- 1.1.2 Riemannian manifold -- 1.1.3 Some formulae and comparison results -- 1.2 Riemannian manifold with boundary -- 1.3 Coupling and applications -- 1.3.1 Transport problem and Wasserstein distance -- 1.3.2 Optimal coupling and optimal map -- 1.3.3 Coupling for stochastic processes -- 1.3.4 Coupling by change of measure -- 1.4 Harnack inequalities and applications -- 1.4.1 Harnack inequality -- 1.4.2 Shift Harnack inequality -- 1.5 Harnack inequality and derivative estimate -- 1.5.1 Harnack inequality and entropy-gradient estimate -- 1.5.2 Harnack inequality and L2-gradient estimate -- 1.5.3 Harnack inequalities and gradient-gradient estimates -- 1.6 Functional inequalities and applications -- 1.6.1 Poincar e type inequality and essential spectrum -- 1.6.2 Exponential decay in the tail norm -- 1.6.3 The F-Sobolev inequality -- 1.6.4 Weak Poincare inequality -- 1.6.5 Equivalence of irreducibility and weak Poincare inequality -- 2. Diffusion Processes on Riemannian Manifolds without Boundary -- 2.1 Brownian motion with drift -- 2.2 Formulae for Pt and RicZ -- 2.3 Equivalent semigroup inequalities for curvature lower bound -- 2.4 Applications of equivalent semigroup inequalities -- 2.5 Transportation-cost inequality -- 2.5.1 From super Poincare to weighted log-Sobolev inequalities -- 2.5.2 From log-Sobolev to transportation-cost inequalities -- 2.5.3 From super Poincare to transportation-cost inequalities -- 2.5.4 Super Poincare inequality by perturbations -- 2.6 Log-Sobolev inequality: Different roles of Ric and Hess -- 2.6.1 Exponential estimate and concentration of -- 2.6.2 Harnack inequality and the log-Sobolev inequality -- 2.6.3 Hypercontractivity and ultracontractivity -- 2.7 Curvature-dimension condition and applications.

2.7.1 Gradient and Harnack inequalities -- 2.7.2 HWI inequalities -- 2.8 Intrinsic ultracontractivity on non-compact manifolds -- 2.8.1 The intrinsic super Poincare inequality -- 2.8.2 Curvature conditions for intrinsic ultracontractivity -- 2.8.3 Some examples -- 3. Reflecting Diffusion Processes on Manifolds with Boundary -- 3.1 Kolmogorov equations and the Neumann problem -- 3.2 Formulae for Pt, RicZ and I -- 3.2.1 Formula for Pt -- 3.2.2 Formulae for RicZ and I -- 3.2.3 Gradient estimates -- 3.3 Equivalent semigroup inequalities for curvature conditionand lower bound of I -- 3.3.1 Equivalent statements for lower bounds of RicZ and I -- 3.3.2 Equivalent inequalities for curvature-dimension condition and lower bound of I -- 3.4 Harnack inequalities for SDEs on Rd and extension to nonconvex manifolds -- 3.4.1 Construction of the coupling -- 3.4.2 Harnack inequality on Rd -- 3.4.3 Extension to manifolds with convex boundary -- 3.4.4 Neumann semigroup on non-convex manifolds -- 3.5 Functional inequalities -- 3.5.1 Estimates for inequality constants on compact manifolds -- 3.5.2 A counterexample for Bakry-Emery criterion -- 3.5.3 Log-Sobolev inequality on locally concave manifolds -- 3.5.4 Log-Sobolev inequality on non-convex manifolds -- 3.6 Modified curvature tensors and applications -- 3.6.1 Equivalent semigroup inequalities for the modified curvature lower bound -- 3.6.2 Applications of Theorem 3.6.1 -- 3.7 Generalized maximum principle and Li-Yau's Harnack inequality -- 3.7.1 A generalized maximum principle -- 3.7.2 Li-Yau type gradient estimate and Harnack inequality -- 3.8 Robin semigroup and applications -- 3.8.1 Characterization of PtQW and D(EQ) -- 3.8.2 Some criteria on Q for (M) = 1 -- 3.8.3 Application to HWI inequality -- 4. Stochastic Analysis on Path Space over Manifolds with Boundary -- 4.1 Multiplicative functional.

4.2 Damped gradient, quasi-invariant ows and integration by parts -- 4.2.1 Damped gradient operator and quasi-invariant flows -- 4.2.2 Integration by parts formula -- 4.3 The log-Sobolev inequality -- 4.3.1 Log-Sobolev inequality on WTx -- 4.3.2 Log-Sobolev inequality on the free path space -- 4.4 Transportation-cost inequalities on path spaces over convex manifolds -- 4.5 Transportation-cost inequality on the path space over nonconvex manifolds -- 4.5.1 The case with a diffusion coefficient -- 4.5.2 Non-convex manifolds -- 5. Subelliptic Diffusion Processes -- 5.1 Functional inequalities -- 5.1.1 Super and weak Poincare inequalities -- 5.1.2 Nash and log-Sobolev inequalities -- 5.1.2.1 Heat kernel estimate -- 5.1.2.2 Nash and log-Sobolev inequalities -- 5.1.3 Gruschin type operator -- 5.1.3.1 Weak Poincare inequality -- 5.1.3.2 Super Poincare inequality -- 5.1.3.3 Nash inequality -- 5.1.3.4 Log-Sobolev inequality -- 5.1.4 Kohn-Laplacian type operator -- 5.1.4.1 Weak Poincare inequality -- 5.1.4.2 Super Poincare inequality -- 5.1.4.3 Nash inequality -- 5.1.4.4 Log-Sobolev inequality -- 5.2 Generalized curvature and applications -- 5.2.1 Derivative inequalities -- 5.2.2 Applications of Theorem 5.2.1 -- 5.2.2.1 L2-derivative estimate and applications -- 5.2.2.2 Entropy-derivative estimate and applications -- 5.2.2.3 Exponential decay and Poincare inequality -- 5.2.2.4 Derivative inequalities by (5.2.3) -- 5.2.3 Examples -- 5.2.4 An extension of Theorem 5.2.1 -- 5.3 Stochastic Hamiltonian system: Coupling method -- 5.3.1 Derivative formulae -- 5.3.2 Gradient estimates -- 5.3.2.1 | Z| is bounded -- 5.3.2.2 | Z| has polynomial growth -- 5.3.2.3 A general case -- 5.3.3 Harnack inequality and applications -- 5.3.3.1 Harnack inequality under (A5.3.1) -- 5.3.3.2 Harnack inequality under assumption (A5.3.2).

5.3.4 Integration by parts formula and shift Harnack inequality -- 5.4 Stochastic Hamiltonian system: Malliavin calculus -- 5.4.1 A general result -- 5.4.2 Explicit formula -- 5.4.3 Two specific cases -- 5.4.3.1 Rank[B0] = m -- 5.4.3.2 A := (1)Z(1) is constant -- 5.5 Gruschin type semigroups -- 5.5.1 Derivative formula -- 5.5.2 Log-Harnack inequality -- 5.5.2.1 Construction of Yt(1) -- 5.5.2.2 Construction of Yt(2) -- Bibliography -- Index.

Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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