## Seminários de probabilidade – 2020

**Coordenação: **Professor Guilherme Ost e Professora Maria Eulalia Vares

Devido à pandemia de coronavírus, as palestras ocorrerão no ambiente virtual gratuito do Google Meet (https://meet.google.com/nxh-optr-wtq) durante os próximos meses. As palestras ocorrerão **às segundas-feiras às 15h**, a menos de algumas exceções devidamente indicadas.

**Lista completa**

In this talk I will try to explain where this equation comes from, why it is interesting, and how its behaviour depends on the spatial dimension. I will mostly focus on the case of dimension 2, and I will comment on a recent result which contradicts a folklore belief from the physics literature.

This is based on joint works with Giuseppe Cannizzaro, Philipp Schönbauer and Fabio Toninelli.

A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity. A crucial step in our analysis is the addition of some non-local events to the multi-scale framework, and interplaying the non-local events with a by now “standard” multi-scale renormalization.

Based on a joint work with Tom Finn (Univ. of Bath).

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.