Semiclassical analysis for diffusions and stochastic processes
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Bibliographic Information
Semiclassical analysis for diffusions and stochastic processes
（Lecture notes in mathematics, 1724）
Springer-Verlag, c2000
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references (p. [329]-345) and index
Description and Table of Contents
Description
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 Levy processes, (iii) complex stochastic Schroedinger 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.
Table of Contents
Gaussian diffusions.- Boundary value problem for Hamiltonian systems.- Semiclassical approximation for regular diffusion.- Invariant degenerate diffusion on cotangent bundles.- Transition probability densities for stable jump-diffusions.- Semiclassical asymptotics for the localised Feller-Courrege processes.- Complex stochastic diffusion or stochastic Schroedinger equation.- Some topics in semiclassical spectral analysis.- Path integration for the Schroedinger, heat and complex diffusion equations.
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