Book chapter
Markov dynamics on the dual object to the infinitedimensional unitary group
We discuss the construction of certain infinitedimensional continuous
time Markov processes, based on the use of intertwined Markov semigroups.
In book
The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U(1), U(2), .... This novel object is a commutative graded algebra R with infinitedimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper. The second part deals with a family of Markov processes on the dual object to the infinitedimensional unitary group U(∞). These processes were defined in a joint work with Alexei Borodin (2012) [5]. The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on R.

We consider a stochastic model of clock synchronization in a wireless network of N sensors interacting with one dedicated accurate time server. For large N we find an estimate of the final time sychronization error for global and relative synchronization. The main results concern the behavior of the network on different timescales tN→∞ , N→∞ . We discuss the existence of phase transitions and find the exact timescales for which an effective clock synchronization of the system takes place.
We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice longtime behavior. We are interested in limit probability laws related to the steady state viewed from the centerofmass coordinate system.
We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice longtime behavior. We are interested in limit probability laws related to the steady state viewed from the centerofmass coordinate system.
This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the upfronts of modern research in linear and nonlinear partial and pseudodifferential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, LandauFokkerPlanck, CahnHilliard, HamiltonJacobiBellman, nonlinear Schroedinger, McKeanVlasov diffusions and their nonlocal extensions, massactionlaw kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and meanfield games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferentialattachment growth on networks. The book is meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudodifferential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.
We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinitedimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process that can be viewed as a collection of simply yet nontrivially correlated twodimensional Gaussian Free Fields. The limiting process has previously arisen via the global scaling limit of spectra for submatrices of Wigner Hermitian random matrices.