Semiclassical Analysis for Diffusions and Stochastic Processes

by Kolokoltsov, Vassili N.

• ISBN13: 9783540669722
• ISBN10: 3540669728
• Format: Paperback
• Copyright: 2000-03-01
• Publisher: Springer Verlag

Summary

This book constitutes the refereed proceedings of the 4th European Conference on Computational Learning Theory, EuroCOLT'99, held in Nordkirchen, Germany in March 1999. The 21 revised full papers presented were selected from a total of 35 submissions; also included are two invited contributions. The book is divided in topical sections on learning from queries and counterexamples, reinforcement learning, online learning and export advice, teaching and learning, inductive inference, and statistical theory of learning and pattern recognition.

Table of Contents

Introduction	1	(16)
Gaussian diffusions
Gaussian diffusion. Probabilistic and analytic approaches.	17	(3)
Classification of Gaussian diffusions by the Young schemes.	20	(5)
Long time behaviour of the Green functions of Gaussian diffusions.	25	(3)
Complex stochastic Gaussian diffusion.	28	(6)
The rate of escape for Gaussian diffusions and scattering for its perturbations.	34	(6)
Boundary value problem for Hamiltonian systems
Rapid course in calculus of variations.	40	(10)
Boundary value problem for non-degenerate Hamiltonians.	50	(9)
Regular degenerate Hamiltonians of the first rank.	59	(13)
General regular Hamiltonians depending quadratically on momenta.	72	(3)
Hamiltonians of exponential growth in momenta.	75	(12)
Complex Hamiltonians and calculus of variations of saddle-points.	87	(5)
Stochastic Hamiltonians.	92	(5)
Semiclassical approximation for regular diffusion
Main ideas of the WKB-method with imaginary phase.	97	(7)
Calculation of the two-point function for regular Hamiltonians.	104	(6)
Asymptotic solutions of the transport equation.	110	(2)
Local asymptotics of the Green function for regular Hamiltonians.	112	(7)
Global small diffusion asymptotics and large deviations.	119	(5)
Asymptotics for non-regular diffusion: an example	124	(4)
Analytic solutions to some linear PDE.	128	(8)
Invariant degenerate diffusion on cotangent bundles
Curvilinear Ornstein-Uhlenbeck process and stochastic geodesic flow.	136	(4)
Small time asymptotics for stochastic geodesic flow.	140	(3)
The trace of the Green function and geometric invariants.	143	(3)
Transition probability densities for stable jump-diffusion
Asymptotic properties of one-dimensional stable laws.	146	(3)
Asymptotic properties of finite-dimensional stable laws.	149	(12)
Transition probability densities for stable jump-diffusion.	161	(17)
Stable jump-diffusions combined with compound Poisson processes.	178	(4)
Stable-like processes.	182	(5)
Applications to the sample path properties of stable jump-diffusions.	187	(4)
Semiclassical asymptotics for the localised Feller-Courrege processes
Maslov's tunnel equations and the Feller-Courrege processes.	191	(3)
Rough local asymptotics and local large deviations.	194	(23)
Refinement and globalisation.	217	(6)
Complex stochastic diffusions or stochastic Schrodinger equations
Semiclassical approximation: formal asymptotics.	223	(6)
Semiclassical approximation: justification and globalisation.	229	(6)
Applications: two-sided estimates to complex heat kernels, large deviation principle, well-posedness of the Cauchy problem.	235	(1)
Path integration and infinite-dimensional saddle-point method.	236	(3)
Some topics in semiclassical spectral analysis.
Double-well splitting.	239	(8)
Low lying eigenvalues of diffusion operators and the life-times of diffusion processes.	247	(5)
Quasi-modes of diffusion operators around a closed orbit of the corresponding classical system.	252	(3)
Path integration for the Schrodinger, heat and complex stochastic diffusion equations
Introduction	255	(8)
Path integral for the Schrodinger equation in p-representation.	263	(4)
Path integral for the Schrodinger equation in x-representation.	267	(2)
Singular potentials.	269	(3)
Semiclassical asymptotics.	272	(4)
Fock space representation.	276	(4)
Appendices
A. Main equation of the theory of continuous quantum measurements.	280	(3)
B. Asymptotics of Laplace integrals with complex phase.	283	(10)
C. Characteristic functions of stable laws.	293	(5)
D. Levy-Khintchine ψ DO and Feller-Courrege processes.	298	(5)
E. Equivalence of convex functions.	303	(2)
F. Unimodality of symmetric stable laws.	305	(7)
G. Infinite divisible complex distributions and complex Markov processes.	312	(10)
H. A review of main approaches to the rigorous construction of path integral.	322	(4)
I. Perspectives and problems.	326	(3)
References	329	(17)
Main notations	346	(1)
Subject Index	347