A functional equation whose unknown is P([0,1]) valued

TL;DR

This paper investigates a functional equation mapping Euclidean space into probability distributions on [0,1], establishing existence, uniqueness, and continuity of solutions, and introduces a new family of distributions generalizing Beta distributions.

Contribution

It introduces a novel functional equation with solutions characterized by probabilistic models, expanding the class of distributions on [0,1].

Findings

Proves existence and uniqueness of solutions.

Characterizes solutions that are diffuse.

Defines a new parametric family of distributions.

Abstract

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.