Limit Theorems for Random Processes

We consider iid birth-and-death processes evolving over timelines indexed by Zd, and a continuous time, time inhomogeneous random walk whose instantaneous jump rate at (x,t) is given by a fixed (exponential) function f of the state of birth-and-death process at (x,t). Other than that, jumps are independent of the birth-and-death environment. The image of f is a subset of (0,1], and has 0 as a limit point. We obtain central limit theorems for the position of the walk when the environment is ergodic under several initial conditions for the environment and jump distributions.

Brownian Aspects of The KPZ Fixed Point
Leandro P. R. Pimentel, Universidade Federal do Rio de Janeiro

The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process on the space of upper-semicontinuous real functions that is conjectured to be at the core of the KPZ universality class. In this talk we will study Brownian aspects of the KPZ fixed point that are related to its local space fluctuations and long time behaviour.

A proof of Sznitman's conjecture about ballistic RWRE
Enrique Guerra, Pontificia Universidad Católica de Chile

We consider a random walk in a uniformly elliptic i.i.d. random environment in Z^d for d>=2. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. To some extent, in order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in [4,5], Sznitman defined the so called conditions (T) and (T'). The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width L. The second one is the requirement that for all \gamma in (0,1) condition (T)_\gamma is satisfied, which in turn is defined as the requirement that the decay is like e^{-CL^\gamma} for some C>0. These conditions in conjunction with renormalization methods have been used to prove ballistic regime and diffusive scaling limit for the random walk process, even in non-independent settings (cf. [2]).

In this talk we will present a recent result in collaboration with A. F. Ram\'{\i}rez [3] that proves a conjecture of Sznitman of 2002 [5], stating that (T) and (T') are equivalent. Hence, this closes the circle proving the equivalence of conditions (T), (T') and (T)_\gamma for some \gamma in (0,1) as conjectured in [5], and also of each of these ballisticity conditions with the polynomial condition (P)_M for M >= 15d+5 introduced in [1].

Bibliography:

1. N. Berger, A. Drewitz and A.F. Ramírez. Effective Polynomial Ballisticity Conditions for Random Walk in Random Environment. Comm. Pure Appl. Math. 67, 1947-1973, (2014).
2. E. Guerra. On the transient $(T)$ condition for random walk in mixing environment. to appear in Ann. Probab. 2019-.
3. E. Guerra and A.F. Ramírez. A proof of Sznitman's conjecture about ballistic RWRE. accepted for publication in Comm. Pure Appl. Math.
4. A.S. Sznitman. On a class of transient random walks in random environment. Ann. Probab. 29, 724-765, (2001).
5. A.S. Sznitman. An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122, 509-544, (2002).

The Tree Builder Random Walk
Rodrigo Ribeiro, Pontificia Universidad Católica de Chile

In this talk, we will discuss the dichotomy recurrence/transience and ballisticity in the context of a random walk that builds its on environment. At each $s$ steps of the walker on the environment, a random number of new vertices is attached to the walker's position. We will present distributional conditions over this random number of new vertices for which we observe distinct sharp behavior on the walker.

This is a joint work with I. Iacobelli, G. Valle and L. Zuaznabar.

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