School of Mathematical Sciences

Vertex Reinforced Jump Process, random Schrödinger operator and hitting time of Brownian motion. menu

Vertex Reinforced Jump Process, random Schrödinger operator and hitting time of Brownian motion.

Speaker: 
Christophe Sabot (Lyon)
Date/Time: 
Wed, 22/03/2017 - 13:00
Room: 
W316

It is well-known that the first hitting time of 0 by a negatively drifted Brownian motion starting at $a>0$ has the inverse Gaussian law. Moreover, conditionally on this first hitting time, the BM up to that time has the law of a 3-dimensional Bessel bridge. In this talk, we will give a generalization of this result to a familly of Brownian motions with interacting drifts. The law of the hitting times will be given by the inverse of the random potential that appears in the context of the self-interacting process called the Vertex Reinforced Jump Process (VRJP). The spectral properties of the associated random Schrödinger operator at ground state are intimately related to the recurrence/transience properties of the VRJP.
We will also explain some "commutativity" property of these BM and its relation with the martingale that appeared in previous work on the VRJP.
Work in progress with Xiaolin Zeng.