The seminar is held online every Wednesday from 16:45 to 17:45 Moscow time.

Permanent link to the Zoom room: http://bit.ly/3HY8K6d 

Room ID: 844 6792 3144 Access code: 697663

October 14, 16:45 msk

Yana Kinderknecht (Butko), HSE Tikhonov Moscow Institute of Electronics and Mathematics, Russia

Physical Origin of Fractional Brownian Motion and Related Processes as Models of Anomalous Diffusion

The most recent and most general family of models of anomalous diffusion is based on the fractional Brownian motion (fBm) with a random diffusion coefficient (superstatistical fBm) together with a time-dependent random Hurst parameter. We provide a dynamical foundation for such general family of models. We consider a dynamical system describing the motion of a test-particle surrounded by N Brownian particles with different masses. This dynamic is governed by underdamped Langevin equations. We prove that, in the limit N→∞, the test-particle diffuses in time according to a quite general (non-Markovian) Gaussian process whose covariance function is determined by the distribution of the masses of the surround-particles. In particular, with proper choices of the distribution of the masses of the surround-particles, we obtain fBm together with a number of other special cases of interest in modeling anomalous diffusion including time-dependent anomalous exponent. Furthermore, when the ensemble heterogeneity of the surround-particles embodying the environment becomes non-uniform and joins with the individual inhomogeneity of the test-particles, we show that, in the limit N→∞, the test-particle diffuses in time according to a quite general conditionally Gaussian process that can be calibrated into a fBm with random diffusion coefficient and random time-dependent Hurst parameter. This is a joint work with Christian Bender, Mirko D'Ovidio and Gianni Pagnini. 

References:

Christian Bender, Yana A. Butko, Mirko D'Ovidio and Gainni Pagnini, Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter, J. Math. Phys. 67, 043301 (2026). https://doi.org/10.1063/5.0292352

https://arxiv.org/abs/2411.18775

https://doi.org/10.17170/kobra-2026040812054

October 7, 16:45 msk

Vassili Kolokoltsov, Lomonosov MSU, NRU HSE, Russia

Mathematical theory of quantum stochastic filtering

The theory of stochastic quantum filtering (or quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author only recently. This new theory led to numerous further developments including the rigorous derivation of filtering equations from the basic principles, the corresponding law of large number limits (propagation of chaos), fractional quantum mechanics of open systems, quantum dynamic and mean-field games. In the talk, we present some basic results and ideas from this story. The presentation will be mostly based on the survey paper Vassili N. Kolokoltsov “Quantum filtering and propagation of chaos for open quantum systems, with applications to quantum feedback control and quantum mean-field games” http://arxiv.org/abs/2607.08507, to appear in UMN.

The recording: YouTube

September 23, 16:45 msk

Sergei Demidov, Lomonosov MSU, Russia

On the Department of the History and Methodology of Mathematics and Mechanics of Moscow State University

V.V. Bobynin began lecturing on the history of pure and applied mathematics at Moscow University as early as the beginning of the 1880s; he also pioneered research in this field in Russia. However, the systematic study of the development of mathematical knowledge and the teaching of a mandatory course on the subject at the University were not initiated until the 1930s by S.A. Yanovskaya and M.Ya. Vygodsky. The Department of the History of Mathematics was established in 1957. K.A. Rybnikov served as its head until his death in 2004. He and I.G. Bashmakova laid the foundations for a new course on the history of mathematics. They also initiated systematic research into the history of the calculus of variations (Rybnikov) and the mathematics of ancient Greece, as well as the theory of divisibility and Diophantine analysis (Bashmakova) — work that was continued by their students, including S.S. Petrova, A.V. Dorofeeva, S.S. Demidov, E.I. Slavutin, G.S. Smirnova, and others. Currently, staff members of the Department are conducting research on the history of mathematics and mechanics in Russia and the USSR (Demidov, Smirnova, V.N. Chinenova, A.N. Bogdanov), on the emergence of ideas regarding the new mechanics and new mathematics in European science during the 15th–17th centuries (E.A. Zaitsev), and on the development of mathematical disciplines from the 18th to the 20th centuries (Demidov, Smirnova, P.N. Antonyuk, Z.A. Kuzicheva, M.A. Podkolzina, Chinenova, Bogdanov). They teach compulsory courses on the history of mathematics and mechanics, special courses on the problems of development of mathematical knowledge, as well as academic mathematics courses at other faculties. At the Cabinet there is a Scientific Research Seminar on the History of Mathematics and Mechanics (founded in 1933).

The recording: YouTube