Title: Mixing rates for processes with long-memory

Speaker: Daniel Yasumasa Takahashi (UFRN)

Our next online seminar will be held next Monday, August 24, from 3 p.m. to 4 p.m. (Rio de Janeiro local time)

The GoogleMeet link for the seminars is HERE. 

Abstract:  Non-Markovian processes are ubiquitous, but they are much less understood compared to Markov processes.  We model non-Markovianity using probability kernels that can depend on its entire history. The continuity rate characterizes how the dependence of kernel on the past decays. One key question is to understand how the mixing rates and decay of correlation are related to the continuity rate. Pollicot (2000) and Bressaud, Fernandez, Galves (1999) showed that if the continuity rate decays as O(1/n^c), for c > 1, then the correlation also decays as O(1/n^c). Johansson, Oberg, Pollicott (2007) proved the uniqueness of the stationary measure compatible with kernels with the continuity rate in O(1/n^c), for c > 1/2. Moreover, Berger, Hoffman, Sidoravicius (2018) established that there are kennels with multiple compatible measures whenever c < 1/2. Therefore, the natural question is to understand the mixing rates and correlation decays when c is in [1/2,1]. In this talk, I will exhibit upper bounds for the mixing rates and correlation decays when the continuity rate decays as  O(1/n^c), for c in (1/2,1].  If time allows, I will show how to apply the result to prove a new weak invariance principle. This talk is based on joint work with Christophe Gallesco.