Large deviations for generalized Polya urns with arbitrary urn function

We consider a generalized two-color Polya urn (black and white balls) first introduced by Hill et al. (1980), where the urn composition evolves as follows: let $\pi :\left[0,1\right]\to \left[0,1\right]$, and denote by $x_n$ the fraction of black balls after step $n$, then at step $n+1$ a black ball is added with probability $\pi \left(x_n\right)$ and a white ball is added with probability $1?\pi \left(x_n\right)$. Originally introduced to mimic attachment under imperfect information, this model has found applications in many fields, ranging from Market Share modeling to polymer physics and biology.In this work we discuss large deviations for a wide class of continuous urn functions $\pi$. In particular, we prove that this process satisfies a Sample-Path Large Deviations principle, also providing a variational representation for the rate function. Then, we derive a variational representation for the limit

$?\left(s\right)=\underset{n\to \infty }{lim}\frac{1}{n}log\mathbb{P}\left(n{x}_n=?sn?\right),s\in \left[0,1\right],$

where $n{x}_n$ is the number of black balls at time $n$, and use it to give some insight on the shape of $?\left(s\right)$. Under suitable assumptions on $\pi$ we are able to identify the optimal trajectory. We also find a non-linear Cauchy problem for the Cumulant Generating Function and provide an explicit analysis for some selected examples. In particular we discuss the linear case, which embeds the Bagchi–Pal Model [6], giving the exact implicit expression for $?$ in terms of the Cumulant Generating Function.     

Multiplicity results for two kinds of equivariant systems

Two equivariant problems of the form $\epsilon \mathrm{\Delta }u=\nabla F\left(u\right)$ are considered, where $F$ is a real function which is invariant under the action of a group $G$, and, using Morse theory, for each problem an arbitrarily great number of orbits of solutions is founded, choosing $\epsilon$ suitably small.The first problem is a $O\left(2\right)$-equivariant system of two equations, which can be seen as a complex Ginzburg-Landau equation, while the second one is a system of $m$ equations which is equivariant for the action of a finite group of real orthogonal matrices $m\times m$.