LABORATORY 16

Laboratory of Stochastic Dynamical Systems

Head of Laboratory – Dr.Sc. (Mathematics), Prof. Aleksandr Veretennikov

Tel.: (095) 299-94-15; E-mail: veretenn@iitp.ru

The leading researchers of the laboratory include:

Dr.Sc. (Math.) V. Kalashnikov
Dr.Sc. (Techn.)	R. Liptser	Dr.	P. Kitsul
Dr.	F. Grigoriev	Dr.	A. Puhalskii
Dr.	O. Gulinsky	Dr.	A. Serebrovskii
Dr.	V. Kistlerov	Dr.	S. Lototsky

Directions of Activity:

· stochastic analysis;

· large deviations;

· mathematical statistics;

· optimal control for moving objects;

· functional programming and language development.

MAIN RESULTS

Poisson equation and diffusion approximation. Three different results are established which turn out to be closely connected so that first one implies the second one which in turn implies the third one. The first result states the smoothness of an invariant diffusion density with respect to the parameter. The second establishes a similar smoothness of the solution of the Poisson equation. The third one states a diffusion approximation result or in other words an averaging of singularity perturbed diffusion for "fully coupled SDE systems". (A. Veretennikov.)

Moderate deviations for smooth processes. A large deviation principle is established for a family of vector-valued smooth random processes defined by a system of ordinary differential equations with perturbations defined by smooth vector function of vector-valued ergodic diffusion. (R. Liptser and A. Veretennikov.)

Large deviations and idempotent probability. The theory of idempotent proba-bility is developed. The results on large deviations are considered from the point of view of weak convergence of the measures corresponding to some stochastic processes to idempotent measures. New results on large deviations for semimartingales are established. The results are applied to queuing systems. (A. Puhalski.)

Some asymptotic problems of quantum mechanics leads to more general setting for large deviations. An approach to large deviations for asymptotic problems without evident probabilistic representation behind is proposed. Some examples provided by the mean field models of quantum statistical mechanics are considered. The approach is based on the Choquet theory of capacities. These tools make it possible to handle non-linear (non-commutative) asymptotic problems as one would handle classical large deviations. (O. Gulinsky.)

New control and synthesis results were obtained for certain classes of nonlinear moving objects. New bounds for optimal control are proposed. (F. Grigoriev.)

Semantics of functional programming languages was studied on the base of FLAC language. (V. Kistlerov.)

Teaching:

- Moscow Institute of Physics and Technology: O. Gulinsky and F. Grigoriev;

- Universities abroad: A. Veretennikov, R. Liptser, A. Puhalskii, P. Kitsul, S. Lototski.

International collaboration. Fruitful collaboration is established with the probability group of the LATP CMI Universite de Provence, Marseille, France, and, in particular, with Professor Etienne Pardoux as its leader.

We also have close contacts with Universite Paris 6 (Professors Jean Jacod and Pierre Priouret); Universite du Main in France (Professor Yuri Kutoyants); Weierstrass Institute for Applied Analysis and Stochastics – WIAS, Berlin, Germany; the Univerity of Warwick, UK (Professor David Elworthy); Mathematical Institute of the University of Copenhagen; University of Trier (Professor Dieter Baum); University of Wuerzburg (Professor Elart von Collani), and some others.

Grants from:

· Russian Foundation for Basic Research (No. 98-01-00062): "Stochastic analysis", head A. Veretennikov.

· INTAS (№ 94-0378): "Stochastic Analysis and Related Topics" (joint project with K. Elworthy, the Warwick University, UK, T. Lyons, Imperial College, London, UK, et al.), co-ordinator A. Veretennikov.

· CNRS/RAS – PICS 99: "Stochastic Analysis", responsible for Moscow team

A. Veretennikov.

Publications in 2001

1. Liptser R., Veretennikov A., and Spokoiny V. Fredlin-Wentzell type moderate deviations for smooth processes (to be published).

Preprint: http://www.mathpreprints.com/math/Preprint/veretenn/20010815/1/ .

2. Veretennikov A. and Pardoux E. On Poisson equation and diffusion approximation // Anal. Prob. (to be published)

Preprint: http://www.mathpreprints.com/math/Preprint/veretenn/20010729/2/ .

3. Puhalskii A. Large deviations and idempotent probability // CRC Press, 2001.

4. Gulinsky O.V. The principle of the largest terms and large deviations for a class of nonlinear functionals with application to quantum statistical mechanics // Proc. of the Workshop on Max-Plus Algebras, Prague, 2001, Elsevier, 2001.

5. Григорьев Ф.Н., Григорьева Е.Н. Количественная взаимосвязь между способностью бурых углей к ожижению и наличием функциональных групп в них // Прикладная химия. 2001, т. 34, вып. 8, с. 1337-1342

6. Григорьев Ф.Н., Кузнецов Н.А. Об управлении движением судна на повороте с учетом ограничений на скорость перекладки руля // XXYlll Всероссийская конференция по управлению движением морскими судами и специальными аппаратами, 2001, 19-22 июня, г. Анапа.

7. Глушков А.В., Григорьев Ф.Н. Управление процессом наблюдения за двумя случайными объектами // IV юбилейная научная конференцию МФТИ, посвященная 50-летию создания Московского физико-технического института, 2001, 23-24 ноября, г. Долгопрудный.

8. Grigoriev F.N., Grigorieva E.N. One estimation method for reliable coefficients In new linear model for complex of physical and chemical coal properties // 6th International Conference on Environment and Mineral Processing, New trends in mineral processing, VSB-TU Ostrava, Czech Rep., 2001, v. 1, p. 15-20.