Semiclassical analysis for diffusions and stochastic processes
The aim of this book is the analytical study of the heat kernel (or Green function) for hypoelliptic diffusions, and more precisely of its semiclassical approximations for small diffusion and/or small time. The study is also extended to diffusions with jumps, to stochastic heat equations, and to complex diffusions. Let us enumerate the main topics which are dealt with. The first chapter considers the case of Gaussian diffusions where exact formulae can be written and the semiclassical expansion is rather explicit. Chapter 2 is devoted to boundary value problems for Hamiltonian systems; after a rapid course on calculus of variations, one studies nondegenerate Hamiltonian systems, then so-called regular degenerate systems; the case of complex and stochastic Hamiltonians is also considered. In Chapter 3, one enters the main subject of the book, namely the WKB method for diffusions which gives a multiplicative expansion for regular Hamiltonian systems with small diffusion; the expansion is first derived formally, then justified. As an application, in Chapter 4, one considers a class of degenerate systems on cotangent bundles. In Chapter 5, a class of diffusions with jumps is introduced; they can be viewed as solutions of equations driven by a symmetric \(\alpha\)-stable process; one also considers diffusions with truncated jumps (this is necessary for the subsequent approximations), and the case where the index \(\alpha\) is not fixed but depends on the position. Semiclassical approximations when these processes have small jumps are derived in Chapter 6. In Chapter 7, one considers the case of complex diffusions. Chapter 8 is devoted to semiclassical spectral analysis, and Chapter 9 gives an approach to a path integral representation for solutions of Schrödinger equations. The book contains also several appendices. The approach which is used is analytical, but diffusions have a well-known stochastic interpretation, and stochastic partial differential equations are also considered, so the book should be useful for both analysts and probabilists. The main applications should be in mathematical physics, but these processes are now used to modelize several other phenomena. The results are precise and are rigorously proved (as a consequence some parts are of course rather technical); most of these results are new. Moreover, the book contains a large list of references and is rather self-contained. It should become a reference monograph on its subject.
stochastic partial differential equations
scientific article; zbMATH DE number 1446986