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Valid Until: 2026-12-31 23:59
Abstract: Interacting Brownian particles are a popular model for various physical systems where a number of particles are subjected to inter-particle forces and ambient noise. We investigate in depth the behaviour of one such model studied earlier both by physicists and mathematicians: a finite chain of Brownian particles, interacting through a pairwise quadratic potential, with one end of the chain fixed and the other end pulled away at slow speed. We study the instant when the chain “breaks”, that is, the distance between two neighboring particles becomes larger than a certain limit. There are three different regimes depending on the relation between the speed of pulling and the Brownian noise. We prove weak limit theorems for the break time and the break position for each regime. The main tools used are weak dependence and Piterbarg-Pickands theorem on Gaussian large deviations.
As a byproduct, for a class of Gaussian stationary processes, we obtain a limit theorem for the last exit time over a slowly growing linear boundary.
The work displays a series of joint works with F.Aurzada, F.Betz, and N.Karagodin.
Bio: Mikhail Lifshits is a professor at his alma mater St. Petersburg State University since 2000 and a board member of the Saint Petersburg Mathematical Society. He got his Master's degree in 1978, his PhD in 1981 from then Leningrad State University and received the Doctor of Science degree in 1993. As an invited professor, M.Lifshits used to teach in France (Strasbourg, Lille, Paris), in the USA (Georgia Tech), in Sweden (Linkoping); he also gave courses in Germany and Finland and many other places. He works in various areas of Random Processes Theory such as Gaussian processes, quantization, small deviations, exit probabilities, etc. as well as in applied Geostatistics. Prof. Lifshits wrote the monograph "Gaussian Random Functions" and textbooks "Lectures on Gaussian Processes" (Springer, 2014) and "Random Processes by Example" (World Scientific, 2015) which are extremely popular among students and young scientists. He is now Editor-in-Chief of the journal Probability Surveys and Associated Editor of the Journal of Theoretical Probability, Theory of Probability and Applications, and some other journals.
