Semiclassical Analysis for Diffusions and Stochastic Processes

Vassili N. Kolokoltsov (auth.)3540669728, 9783540669722

The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.

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Table of contents :
Introduction….Pages 1-16
Gaussian diffusions….Pages 17-39
Boundary value problem for Hamiltonian systems….Pages 40-96
Semiclassical approximation for regular diffusion….Pages 97-135
Invariant degenerate diffusion on cotangent bundles….Pages 136-145
Transition probability densities for stable jump-diffusions….Pages 146-190
Semiclassical asymptotics for the localised Feller-Courrège processes….Pages 191-222
Complex stochastic diffusion or stochastic Schrödinger equation….Pages 223-238
Some topics in semiclassical spectral analysis….Pages 239-254
Path integration for the Schrödinger, heat and complex diffusion equations….Pages 255-279