# A simple method to find all solutions to the functional equation of the smoothing transform

## TL;DR

This paper introduces a simplified, robust method for solving the functional equation of the smoothing transform, applicable under weaker conditions and potentially extendable to more general stochastic setups.

## Contribution

It presents a new three-step approach to characterize solutions of the smoothing transform's functional equation, improving on earlier methods with broader applicability.

## Key findings

- Provides a complete characterization of solutions as Laplace transforms.
- Establishes a correspondence between solutions and fractal random measures.
- Works under weaker boundary conditions than previous approaches.

## Abstract

Given a nonincreasing null sequence $T = (T_j)_{j \ge 1}$ of nonnegative random variables satisfying some classical integrability assumptions and $\mathbb{E}(\sum_{j}T_{j}^{\alpha})=1$ for some $\alpha>0$, we characterize the solutions of the well-known functional equation $$ f(t)\,=\,\textstyle\mathbb{E}\left(\prod_{j\ge 1}f(tT_{j})\right),\quad t\ge 0, $$ related to the so-called smoothing transform and its min-type variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new three-step method whose merits are that (1) it simplifies earlier approaches in some relevant aspects, (2) it works under weaker, close to optimal conditions in the so-called boundary case when $\mathbb{E}\big(\sum_{j\ge 1}T_{j}^{\alpha}\log T_{j}\big)=0$, (3) it can be expected to work as well in more general setups like random environment. At the end of this article, we also give a one-to-one correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk.