Limit theorems for the one dimensional random walk with random resetting to the maximum
Báo cáo viên: Nguyễn Văn Quyết (Viện Toán học)

Thời gian: 14h Thứ 5, ngày 25/03/2021

Địa điểm: P507 nhà A6 hoặc online qua link


Tóm tắt: The first part of this talk aims to study a model of one-dimensional random walk with memory to the maximum position described as follows. At each step the walker resets to the rightmost visited site with probability r ∈ (0, 1) and moves as the simple random walk with remaining probability. Using the approach of renewal theory, we prove the laws of large numbers and the central limit theorems for the random walk. These results reprove and significantly enhance the analysis of asymptotic behavior of the mean value and variance of the process established by Majumdar, Sabhapandit and Schehr.

In the second part of this talk, we expand the analysis to the situation when the resetting rate to the maximum position decreases over time by assuming that at the step n the resetting probability is $r_n = min{r.n^{−a},1/2}$ with r, a positive parameters. For this model, we first establish the asymptotic behavior of the mean values of $X_n$ -the current position and $M_n$-the maximum position of the random walk. As a consequence, we observe an interesting phase transition in the asymptotic behavior of E[Xn]/E[Mn] when a varies. Finally, when a > 1, we provide the limit theorem for X_n and M_n. This is in joint work with my supervisors Can Van Hao and Doan Thai Son.