Semiclassical Analysis for Diffusions and Stochastic Processes

by Vassili N. Kolokoltsov

Overview

The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and shastic 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 shastic 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.

Product Details

ISBN-13:	9783540669722
Publisher:	Springer Berlin Heidelberg
Publication date:	04/26/2000
Series:	Lecture Notes in Mathematics , #1724
Edition description:	2000
Pages:	356
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

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-Courrège processes.- Complex shastic diffusion or shastic Schrödinger equation.- Some topics in semiclassical spectral analysis.- Path integration for the Schrödinger, heat and complex diffusion equations.