This is the page of the spring semester of 2022 of the United Seminar of the Department of Probability Theory of the Faculty of Mechanics and Mathematics of Moscow State University. The permanent website of the seminar is here. The seminar is a continuation of the research seminar of the Department of Probability Theory under the leadership of A.N. Kolmogorov and B.V. Gnedenko.

The list of reports from 2020 and previous years can be viewed at this link.
The list of reports from 2021 can be viewed here and here.

Previous reports

February 16, 16:45 MSK
Luigi Accardi, Universitá di Roma Torvergata, Roma Centro Interdipartimentale Vito Volterra

Deduction of quantum theory and its extensions from classical probability

For more than 30 years quantum probability (QP) has been considered as a generalization of classical probability (CP). Now we understand that this is not true: QP is a deeper level of CP. Main goal of the talk is to explain the statement above and why, now and in the future, the best way to introduce people to quantum me¬chanics and its natural extensions is a course in classical probability. If time permits, some feed-back of this new approach, for classical probability, will be illustrated through some examples.

A recording of the report can be found at this link.

Date is TBA
Georgii Riabov, Institute of Mathematics of Ukrainian Academy of Sciences

Coalescing stochastic flows

The talk is devoted to properties of coalescing stochastic flows, i.e. families of random mappings of the phase space that satisfy the evolutionary property, act independently on disjoint time intervals and possess stationary distributions. A flow is called coalescing if mappings that constitute this flow are not injective with positive probability. Under wide assumptions on distributions of n-point motions on the metric graph, the existence of the corresponding stochastic flow will be proved. Also, we will discuss dual flows for coalescing stochastic flows on the real line.

March 9, 16:45 MSK
Aleksandr Veretennikov, IITP RAS, MSU, NRU HSE

On McKean–Vlasov equations

A review of results on existence and (strong and weak) uniqueness of McKean - Vlasov equations will be presented. It will be also shown why they are interesting (in particular, because the measures of the solutions satisfy nonlinear Fokker - Planck - Kolmogorov equations) and how they relate to "multiparticle ensembles": this theory is called "chaos expansion". A separate question about ergodic properties of solutions of such equations will be touched if time allows.

March 23, 16:45 MSK
Liubov Markovich, Delft University of Technology, Trapeznikov Institute of Control Sciences, Institute for information transmission problems, Russian Quantum Center

Quantum enhanced measurement of many-body observables

In different areas of science the problem of finding the expectation value of an operator corresponding to an observable of a system is of utmost importance. For example, in many tasks in condensed matter physics, materials science, quantum chemistry and combinatorial optimization, the goal is to find spectral properties, the ground state energy or the lowest eigenvalue of a Hamiltonian. Direct estimation of the expectation value of observable decomposed into weighted sum of N Pauli strings is not straight forward and for complex system, it deemed to be nearly impossible.

In this project we propose an alternative approach to the current method of individually measuring each Pauli string, with a further classical summation of all values (we call this approach classical method). Our idea is to sum all the Pauli strings coherently. Using the phase kickback method in quantum phase estimation, each Pauli string is encoded in the phase and written into an ancilla qubit in such a way that the sum of all Pauli strings is encoded in one phase as a sum of nonlinear functions. Our circuit contains two parts: a target quantum system with a short coherent time qubits and a measurement device with a memory ancilla qubit that has a long coherent time enough to encode each Pauli string and to proceed the QPE. As a result, our approach promises linear improvement in N comparing to the classical one.

This is joint work with J. Borregard and A. Almasi.

A recording of the report can be found at this link.

April 13, 16:45 MSK
Victor Vedenyapin, MSU

On Vlasov Type Equations

Now there are Vlasov-Poisson equations, Vlasov-Maxwell equations, Vlasov-Einstein equations, the names were introduced mainly by French mathematicians (Choquet-Brua, etc.), but have become generally accepted. The story of Vlasov-type equations will be presented. In classical textbooks (Pauli; Fock,; Landau and Lifshitz; Dubrovin, Novikov, Fomenko; Weinberg; Vlasov ...), equations for fields in the Einstein and Maxwell equations are proposed without deducing the right parts. Here we give the derivation of the right-hand sides of the Maxwell and Einstein equations within the framework of the Vlasov-Maxwell-Einstein equations from the classical, but slightly more general principle of least action. A method of transition from kinetic equations to hydrodynamic consequences is proposed, as it was done earlier by A.A.Vlasov himself. In the case of Hamiltonian mechanics, a transition from the hydrodynamic consequences of the Liouville equation to the Hamilton-Jacobi equation is possible. Thus, in the non-relativistic case, Milne-McCree solutions are obtained, a non-relativistic analogue of Friedman-type solutions of the evolution of the Universe.

A recording of the report can be found at this link.