Random walks with excursion sign memory, associated counting problems, and convergence to the skew Brownian motion

Random walks with excursion sign memory, associated counting problems, and convergence to the skew Brownian motion

Thời gian: 09:00 đến 11:00 Ngày 11/08/2017

Địa điểm: C2-714, VIASM

Báo cáo viên: Nabil KAZI-TANI

Tóm tắt:
We are interested in a discrete random walk on integers that take the steps (1, +1) and (1, −1) with equal probability, when it is not equal to 0. When it reaches the state 0, the behavior of the walk depends on wether it touched the x-axis coming from a positive or a negative excursion. The excursions of this walk do not form an i.i.d. sequence. We will give an explicit expression for the one dimensional distribution of this walk. As a by-product, we obtain a new simple combinatorial interpretation of k-fold convolutions of Catalan numbers.

We will discuss the continuous time limit of such discrete random walks and explain their convergence to the skew Brownian motion. 

We will also explain how to generalize these walks, either to the case of excursions around a countable number of levels (compared to one level at 0) or to random walks on trees, and their convergence in the continuous time limit, to the multi-skewed Brownian motion.

This is a joint work with Dylan Possamaï (Columbia University).