ISBN: 3-540-66972-8
TITLE: Semiclassical Analysis for Diffusions and Stochastic Processes
AUTHOR: Kolokoltsov, Vassili N.
TOC:

Introduction 1
Chapter 1. Gaussian diffusions
1. Gaussian diffusion. Probabilistic and analytic approaches 17
2. Classification of Gaussian diffusions by the Young schemes 20
3. Long time behaviour of the Green functions of Gaussian diffusions. 25
4. Complex stochastic Gaussian diffusion 28
5. The rate of escape for Gaussian diffusions and scattering for its perturbations3 4
Chapter 2. Boundary value problem for Hamiltonian systems
1. Rapid course in calculus of variations. 40
2. Boundary value problem for non-degenerate Hamiltonians. 50
3. Regular degenerate Hamiltonians of the first rank. 59
4. General regular Hamiltonians depending quadratically on momenta 72
5. Hamiltonians of exponential growth in momenta 75
6. Complex Hamiltonians and calculus of variations of saddle-points 87
7. Stochastic Hamiltonians 92
Chapter 3. Semiclassical approximation for regular diffusion
1. Main ideas of the WKB-method with imaginary phase. 97
2. Calculation of the two-point function for regular Hamiltonians. 104
3. Asymptotic solutions of the transport equation 110
4. Local asymptotics of the Green function for regular Hamiltonians 112
5. Global small diffusion asymptotics and large deviations 119
6. Asymptotics for non-regular diffusion: an example 124
7. Analytic solutions to some linear PDE 128
Chapter 4. Invariant degenerate diffusion on cotangent bundles
1. Curvilinear Ornstein-Uhlenbeck process and stochastic geodesic flow. 136
2. Small time asymptotics for stochastic geodesic flow 140
3. The trace of the Green function and geometric invariants 143
Chapter 5. Transition probability densities for stable jump-diffusion
1. Asymptotic properties of one-dimensional stable laws 146
2. Asymptotic properties of finite-dimensional stable laws 149
3. Transition probability densities for stable jump-diffusion. 161
4. Stable jump-diffusions combined with compound Poisson processes. 178
5. Stable-like processes. 182
6. Applications to the sample path properties of stable jump-diffusions. 187
Chapter 6. Semiclassical asymptotics for the localised Feller-Courrge processes
1. Maslov's tunnel equations and the Feller-Courrge processes. 191
2. Rough local asymptotics and local large deviations 194
3 Refinement and globalisation. 217
Chapter 7. Complex stochastic diffusions or stochastic Schrdinger equations
1. Semiclassical approximation: formal asymptotics 223
2. Semiclassical approximation: justification and globalisation 229
3. Applications: two-sided estimates to complex heat kernels, large deviation principle, well-posedness of the Cauchy problem 235
4. Path integration and infinite-dimensional saddle-point method. 236
Chapter 8. Some topics in Semiclassical spectral analysis
1. Double-well splitting. 239
2. Low lying eigenvalues of diffusion operators and the life-times of diffusion processes. 247
3. Quasi-modes of diffusion operators around a closed orbit of the corresponding classical system 252
Chapter 9. Path integration for the Schrdinger, heat and complex stochastic diffusion equations
1. Introduction. 255
2. Path integral for the Schrdinger equation in p-representation. 263
3. Path integral for the Schrdinger equation in x-representation. 267
4. Singular potentials. 269
5. Semiclassical asymptotics 272
6. Fock space representation 276
Appendices
A. Main equation of the theory of continuous quantum measurements 280
B. Asymptotics of Laplace integrals with complex phase. 283
C. Characteristic functions of stable laws 293
D. Lvy-Khintchine DO and Feller-Courrge processes. 298
E. Equivalence of convex functions 303
F. Unimodality of symmetric stable laws 305
G. Infinite divisible complex distributions and complex Markov processes. 312
H. A review of main approaches to the rigorous construction of path integral. 322
I. Perspectives and problems 326
References 329
Main notations 346
Subject Index 347
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