Diffusions of Multiplicative Cascades

TL;DR

This paper introduces a continuous-time, measure-valued process for multiplicative cascades, extending their classical construction with independent increment processes, and explores applications in tree polymers and random geometry.

Contribution

It develops a novel continuous-time, measure-valued cascade process with Markov and martingale properties, enhancing the modeling of complex hierarchical systems.

Findings

The process is Markovian and martingale.

Under certain conditions, the process is continuous.

Applications include models of tree polymers and 1D random geometry.

Abstract

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. We discuss applications of this process to models of tree polymers and one-dimensional random geometry.