## Seminários de probabilidade – 2019

**Coordenação: **Professora Maria Eulalia Vares

As palestras ocorrerem na sala B106-b nas segundas-feiras às 15h30, a menos de algumas exceções devidamente indicadas.

**Lista completa**

Authors: Zanini, C. T. P, Paulon, G., Mueller, P.

We propose the notion of replicas in the context of discrete choices and introduce axioms that support which we call the Luce model with replicas. Unlike other relations proposed in the literature that can deal with the duplicates problem, ours entails replicas as a combination of duplicates and stochastically perfect substitutes, which induces a partition of the entire set of alternatives into endogeneous nests of replicas. Our model is less restrictive than Luce.s model and more parsimonious than the available models that may deal with the violation of the constant-ratio rule anticipated by Debreu (1960).

$$

\partial X = -(-\Delta)^{1/2} X – \sinh(\gamma X) + \xi,

$$

where $(-\Delta)^{1/2}$ is the half-laplacian, and $\xi$ is the space-time white noise. As the solution is not point-wise welldefined function, we will have to define the meaning of $\sinh(\gamma X)$. We will also discuss the basic ideas between da Pratto Debusche approach to non-linear SPDE’s and more modern techniques, such as regularity structures.

It turns out that $L_c(p)$ is polynomial if $r le a_3$, exponential if $a_3 < r le a_2 + a_3$, doubly exponential if $a_2 + a_3 < r le a_1 + a_2 + a_3$, and infinite if $r > a_1 + a_2 + a_3$. In this talk we will focus on the case $r = a_3 + 1$, and show how to determine $log L_c(p)$ up to a constant factor. The main new tool, which we call the {it beams process}, allows one to reduce the problem to proving an exponential decay property for a certain two-dimensional model whose behaviour resembles site percolation.