Semiclassical analysis for diffusions and stochastic processes / Vassili N. Kolokoltsov

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 α-stable process; one also considers diffusions with truncated jumps (this is necessary for the subsequent approximations), and the case where the index α 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. ... (Zentralblatt)