Table of Contents
Preface v
1 Preliminaries 1
1.1 Riemannian manifold 1
1.1.1 Differentiable manifold 1
1.1.2 Riemannian manifold 3
1.1.3 Some formulae and comparison results 9
1.2 Riemannian manifold with boundary 11
1.3 Coupling and applications 15
1.3.1 Transport problem and Wasserstein distance 16
1.3.2 Optimal coupling and optimal map 18
1.3.3 Coupling for stochastic processes 19
1.3.4 Coupling by change of measure 22
1.4 Harnack inequalities and applications 24
1.4.1 Harnack inequality 24
1.4.2 Shift Harnack inequality 31
1.5 Harnack inequality and derivative estimate 33
1.5.1 Harnack inequality and entropy-gradient estimate 33
1.5.2 Harnack inequality and L2-gradient estimate 36
1.5.3 Harnack inequalities and gradient-gradient estimates 37
1.6 Functional inequalities and applications 39
1.6.1 Poincaré type inequality and essential spectrum 39
1.6.2 Exponential decay in the tail norm 42
1.6.3 The F-Sobolev inequality 42
1.6.4 Weak Poincaré inequality 43
1.6.5 Equivalence of irreducibility and weak Poincaré inequality 45
2 Diffusion Processes on Riemannian Manifolds without Boundary 49
2.1 Brownian motion with drift 49
2.2 Formulae for ∇Pt and Ricz 54
2.3 Equivalent semigroup inequalities for curvature lower bound 60
2.4 Applications of equivalent semigroup inequalities 72
2.5 Transportation-cost inequality 77
2.5.1 From super Poincaré to weighted log-Sobolev inequalities 79
2.5.2 From log-Sobolev to transportation-cost inequalities 82
2.5.3 From super Poincaré to transportation-cost inequalities 87
2.5.4 Super Poincaré inequality by perturbations 92
2.6 Log-Sobolev inequality: Different roles of Ric and Hess 95
2.6.1 Exponential estimate and concentration of μ 96
2.6.2 Harnack inequality and the log-Sobolev inequality 98
2.6.3 Hypercontractivity and ultracontractivity 102
2.7 Curvature-dimension condition and applications 109
2.7.1 Gradient and Harnack inequalities 109
2.7.2 HWI inequalities 120
2.8 Intrinsic ultracontractivity on non-compact manifolds 127
2.8.1 The intrinsic super Poincaré inequality 129
2.8.2 Curvature conditions for intrinsic ultracontractivity 131
2.8.3 Some examples 136
3 Reflecting Diffusion Processes on Manifolds with Boundary 141
3.1 Kolmogorov equations and the Neumann problem 142
3.2 Formulae for ∇Pt,Ricz and I 146
3.2.1 Formula for ∇Pt 146
3.2.2 Formulae for Ricz and I 149
3.2.3 Gradient estimates 152
3.3 Equivalent semigroup inequalities for curvature condition and lower bound of I 159
3.3.1 Equivalent statements for lower bounds of Ricz and I 159
3.3.2 Equivalent inequalities for curvature-dimension condition and lower bound of I 165
3.4 Harnack inequalities for SDEs on Rd and extension to non- convex manifolds 167
3.4.1 Construction of the coupling 169
3.4.2 Harnack inequality on Rd 174
3.4.3 Extension to manifolds with convex boundary 176
3.4.4 Neumann semigroup on non-convex manifolds 180
3.5 Functional inequalities 181
3.5.1 Estimates for inequality constants on compact manifolds 181
3.5.2 A counterexample for Bakry-Emery criterion 184
3.5.3 Log-Sobolev inequality on locally concave manifolds 186
3.5.4 Log-Sobolev inequality on non-convex manifolds 190
3.6 Modified curvature tensors and applications 195
3.6.1 Equivalent semigroup inequalities for the modified curvature lower bound 196
3.6.2 Applications of Theorem 3.6.1 200
3.7 Generalized maximum principle and Li-Yau's Harnack inequality 204
3.7.1 A generalized maximum principle 206
3.7.2 Li-Yau type gradient estimate and Harnack inequality 210
3.8 Robin semigroup and applications 214
3.8.1 Characterization of PtQ,W and D(εQ) 215
3.8.2 Some criteria on λQ for μ(M) = 1 218
3.8.3 Application to HWI inequality 222
4 Stochastic Analysis on Path Space over Manifolds with Boundary 227
4.1 Multiplicative functional 228
4.2 Damped gradient, quasi-invariant flows and integration by parts 234
4.2.1 Damped gradient operator and quasi-invariant flows 234
4.2.2 Integration by parts formula 236
4.3 The log-Sobolev inequality 239
4.3.1 Log-Sobolev inequality on WxT 239
4.3.2 Log-Sobolev inequality on the free path space 242
4.4 Transportation-cost inequalities on path spaces over convex manifolds 245
4.5 Transportation-cost inequality on the path space over non-convex manifolds 251
4.5.1 The case with a diffusion coefficient 252
4.5.2 Non-convex manifolds 255
5 Subelliptic Diffusion Processes 257
5.1 Functional inequalities 259
5.1.1 Super and weak Poincaré inequalities 259
5.1.2 Nash and log-Sobolev inequalities 265
5.1.3 Gruschin type operator 275
5.1.4 Kohn-Laplacian type operator 277
5.2 Generalized curvature and applications 283
5.2.1 Derivative inequalities 285
5.2.2 Applications of Theorem 5.2.1 289
5.2.3 Examples 293
5.2.4 An extension of Theorem 5.2.1 300
5.3 Stochastic Hamiltonian system: Coupling method 303
5.3.1 Derivative formulae 304
5.3.2 Gradient estimates 309
5.3.3 Harnack inequality and applications 317
5.3.4 Integration by parts formula and shift Harnack inequality 322
5.4 Stochastic Hamiltonian system: Malliavin calculus 328
5.4.1 A general result 328
5.4.2 Explicit formula 332
5.4.3 Two specific cases 338
5.5 Gruschin type semigroups 343
5.5.1 Derivative formula 343
5.5.2 Log-Harnack inequality 353
Bibliography 365
Index 377
