Yuri Kondratiev

HARMONIC ANALYSIS ON CONFIGURATION SPACE I: GENERAL THEORY

We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.

Analysis on Poisson and Gamma Spaces

We study the spaces of Poisson, compound Poisson and Gamma noises as special cases of a general approach to non-Gaussian white noise calculus, see Ref. 18. We use a known unitary isomorphism between Poisson and compound Poisson spaces in order to transport analytic structures from Poisson space to compound Poisson space. Finally we study a Fock type structure of chaos decomposition on Gamma space.

Wick calculus in Gaussian analysis

We define an extension of the distribution spaces conventionally used in Gaussian analysis. This space, characterized by analytic properties of S-transforms, allows for a calculus based on the Wick product. We indicate some of its features.

CORRELATION FUNCTIONS AND INVARIANT MEASURES IN CONTINUOUS CONTACT MODEL

We study the continuous version of the contact model. Using an analytic approach, we construct the non-equilibrium contact process as a Markov process on configuration space. The construction is based on the analysis of correlation functions evolution. The problem concerning invariant measures as well as asymptotics of correlation functions are also studied.

Markov evolutions and hierarchical equations in the continuum. I: one-component systems

General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.

ON CONTACT PROCESSES IN CONTINUUM

We introduce a continuous version of the contact model which is well known and widely studied in the lattice case. Under certain general assumptions on the infection spreading characteristics, we construct the contact process as a Markov process in the configuration space of the system.

The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach

Quantum statistical mechanics plays a major role in many fields such as, for instance, thermodynamics, plasma physics, solid-state physics, and the study of stellar structure. While the theory of quantum harmonic oscillators is relatively simple, the case of anharmonic oscillators, a mathematical model of a localized quantum particle, is more complex and challenging. Moreover, infinite systems of interacting quantum anharmonic oscillators possess interesting ordering properties with respect to quantum stabilization. This book presents a rigorous approach to the statistical mechanics of such systems, in particular with respect to their actions on a crystal lattice. The text is addressed to both mathematicians and physicists, especially those who are concerned with the rigorous mathematical background of their results and the kind of problems that arise in quantum statistical mechanics. The reader will find here a concise collection of facts, concepts, and tools relevant for the application of path integrals and other methods based on measure and integration theory to problems of quantum physics, in particular the latest results in the mathematical theory of quantum anharmonic crystals. The methods developed in the book are also applicable to other problems involving infinitely many variables, for example, in biology and economics.

Nonequilibrium Glauber-type dynamics in continuum

We construct the nonequilibrium Glauber dynamics as a Markov process in configuration space for an infinite particle system in continuum with a general class of initial distributions. This class we define in terms of correlation functions bounds and it is preserved during the Markov evolution we constructed.

Vlasov Scaling for Stochastic Dynamics of Continuous Systems

We describe a general derivation scheme for the Vlasov-type equations for Markov evolutions of particle systems in continuum. This scheme is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations. Several examples of realization of the proposed approach in particular models are presented.

ONE-PARTICLE SUBSPACE OF THE GLAUBER DYNAMICS GENERATOR FOR CONTINUOUS PARTICLE SYSTEMS

We consider a Glauber-type stochastic dynamics of continuous particle systems in ℝd. We construct a one-particle invariant subspace of the generator of this dynamics in the high temperature and low density regime. We prove that under some additional assumptions on the decay of the potential the restriction of the generator on the one-particle subspace is unitary equivalent to the operator of the multiplication by a bounded smooth real-valued function. As a consequence we estimate the spectral gap of the generator and find the second gap between the one-particle branch and the rest of the spectrum.