Diffusions of multiplicative cascades

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Alberts, Tom¹  Rifkind, Ben² 

[1] CALTECH, Dept Math, Pasadena, CA 91125 USA

[2] Univ Toronto, Dept Math, Toronto, ON M5S 2E4, Canada

关键词: Measure-valued Markov process;    Infinite dimensional stochastic calculus;    Tree polymers;   

DOI  :  10.1016/j.spa.2013.10.007

来源: Elsevier

【 摘 要 】

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an lid 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. For Gaussian independent increment processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Holder exponent in the Wasserstein distance on measures. We also discuss applications of this process to the model of tree polymers. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free