报告学者：Sabir Umarov

报告者单位：University of New Haven

报告时间：2025/07/05  14:10-14:45

报告地点：腾讯会议 699-622-572

Abstract: The classical Fokker-Planck (FP) equation plays an important role in modern sciences describing deterministically random processes driven by Wiener process (Brownian motion). The Fokker-Planck equation is a parabolic partial differential equation expressing time evolution of the corresponding stochastic process. Stochastic processes accompanying with small and large jumps can be modeled by SDEs driven by Levy processes. Their associated FP equations are pseudo-differential equations. The symbols of these pseudo-differential operators are nonregular and cannot be described with the help of classical theory of pseudo-differential operators.

Levy and Wiener processes can be interpreted as continuous time random walk (CTRW) limits. At the same time, in general, CTRW limit processes form a wide class of stochastic processes, particularly containing Levy and Wiener processes. SDEs driven by CTRW limit processes allow to model many random processes arising in natural, social, and formal sciences. Examples include protein movement in cell membrane, underground water flows, financial transactions in insurance companies, currency rate exchanges, motions in fractal media, and many more.

My talk is devoted to the general theory of CTRW-limits processes in the Skorokhod cadlag spaces and SDEs driven by such general driving processes. We will establish associated FP equations, as well. Since CTRW limit processes are represented as a time changed processes subordinated to an inverse of a mixture of Levy subordinators, the FP equations emerge as fractional order pseudo-differential equations. Therefore, we also introduce a brief survey of related fractional calculus of utilized fractional order differential operators. 

报告学者简介：Professor Umarov is PHD and Dr. of Sciences obtained from Moscow Energy Research Institute (Technical University). He worked at different times for the National University of Uzbekistan, University of New Mexico (USA), Tufts University (USA), and currently works for the University of New Haven (USA).

Professor Umarov’s research interests mainly focused on the study of mathematical models of various kind of evolution processes through the partial, stochastic, pseudo-, integro, operator-, functional-, and fractional order differential equations. Study of solution spaces, properties of solutions, tools and methods of solution, numerical approximations, simulation techniques and related aspects are an essential part of his research. Applications of these models to cell biology, anomalous diffusion, financial mathematics, and other fields are also of special interest.

Professor Umarov has published over 100 scientific papers and authored the following books:  

Introduction to fractional and pseudodifferential equations with singular symbols, Springer, 2025.

Beyond the triangle: Brownian motion, Ito calculus, and Fokker-Planck equation – fractional generalizations, World Scientific, 2018 (co-authors M. Hahn, K. Kobayashi).

Mathematical foundations of nonextensive statistical mechanics, World Scientific, 2022 (co-author C. Tsallis).