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Principle Seminar of the Department of Probability Theory, Moscow State University
March 3, 2021 16:45–17:40, Moscow, MSU, auditorium 12-24
 


Uniqueness problems for the Fokker-Planck-Kolmogorov equations

V. I. Bogachev

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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Abstract: This talk concerns several elementary formulated uniqueness problems connected with the stationary Kolmogorov equation and the Cauchy problem for the evolution Fokker  Planck  Kolmogorov equation. Some of these problems, posed by Kolmogorov himself in the 1930s or naturally connected with Kolmogorov's questions, have been solved relatively recently, but others remain open. We shall mainly discuss the case of the unit diffusion coefficient and a smooth drift b, where the stationary equation with respect to a measure has the form ∆ − div(b) = 0 or the same form for the solution density, and the parabolic equation for the density p(x,t) is written as ∂_t p = ∆p − div(bp). Such equations can be considered in the class of probability measures as well as in the class of bounded (possibly, signed) measures, which leads to different interesting problems. The uniqueness problems we discuss are also related to stationary measures of diffusions, Chapman  Kolmogorov equations and semigroups generated by second order elliptic operators. For understanding the essence of our main problems it is enough to have acquaintance with two semesters of calculus.

Website: https://youtu.be/MyWa23nWU8E

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