Table of contents

CONTENTS Introduction Chapter I. Distributions of functionals of Brownian local time § 1. Brownian local time § 2. Distributions of Brownian local time § 3. The Markov property § 4. Distributions of integral functionals of Brownian local time and functionals of supremum type § 5. Distributions of functionals of occupation time type § 6. Distribution of the supremum of the increments of Brownian local time Chapter II. Properties of trajectories of Brownian local time § 1. Continuity § 2. Law of the iterated logarithm § 3. Brownian local time - a quasi-martingale Chapter III. Convergence of stochastic processes to Brownian local time § 1. Brownian local time as a limit process § 2. Hausdorff measure of the level sets of Brownian motion § 3. On the character of the convergence to Brownian local time § 4. Limit theorems for the occupation times of Brownian motion § 5. On estimation of Brownian local time from observations of Brownian motion at discrete instants Chapter IV. Invariance principle for local times § 1. Weak invariance principle for local times § 2. Strong invariance principle for local times § 3. Limit behaviour of the differences between Brownian local time and processes converging to it § 4. Law of the iterated logarithm for local time of a random walk Chapter V. Applications of Brownian local time § 1. The generalized Itô formula § 2. Non-negative continuous homogeneous additive functionals of Brownian motion References

CONTENTS Introduction § 1. Normal form of a linear Hamiltonian system with periodic coefficients § 2. Normal form in the neighbourhood of a periodic solution § 3. Neighbourhood of an invariant torus § 4. A system with two degrees of freedom § 5. Remarks References

CONTENTS Introduction § 1. Asymptotic expansion of Laplace's variational integrals § 2. Computation of dispersive media § 3. Extremal property of the hexagonal distribution of discs § 4. Random chess structure § 5. Asymptotic expansion of the effective conductivity for a small concentration of the non-conducting cells Concluding remarks References

CONTENTS Introduction Chapter I. The spectral theory of the non-stationary Schrödinger operator § 1. The perturbation theory for formal Bloch solutions § 2. The structure of the Riemann surface of Bloch functions § 3. The approximation theorem § 4. The spectral theory of finite-gap non-stationary Schrödinger operators § 5. The completeness theorem for products of Bloch functions Chapter II. The periodic problem for equations of Kadomtsev-Petviashvili type § 1. Necessary information on finite-gap solutions § 2. The perturbation theory for finite-gap solutions of the Kadomtsev-Petviashvili –2 equation § 3. Whitham equations for space two-dimensional "integrable systems" § 4. The construction of exact solutions of Whitham equations § 5. The quasi-classical limit of two-dimensional integrable equations. The Khokhlov-Zabolotskaya equation Chapter III. The spectral theory of the two-dimensional periodic Schrödinger operator for one energy level § 1. The perturbation theory for formal Bloch solutions § 2. The structure of complex "Fermi-curves" § 3. The spectral theory of "finite-gap operators with respect to the level E₀" and two-dimensional periodic Schrödinger operators References

CONTENTS Introduction § 1. Preliminary information § 2. Construction of particular solutions for a difference equation of hypergeometric type on non-uniform lattices § 3. Some properties of difference functions of hypergeometric type § 4. Classical orthogonal polynomials of a discrete variable on non-uniform lattices § 5. Functions of a discrete variable of the second kind § 6. Some solutions of a difference equation of hypergeometric type on non-uniform lattices Conclusion Appendix. A transformation of the original equation References