Diffusions of multiplicative cascades
- Creators
- Alberts, Tom
- Rifkind, Ben
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. 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 Hölder exponent in the Wasserstein distance on measures. We also discuss applications of this process to the model of tree polymers.
Additional Information
© 2013 Elsevier B.V. Received 26 May 2013; received in revised form 29 September 2013; accepted 27 October 2013; Available online 4 November 2013. We thank Bálint Virág for several helpful comments and suggestions, and Sourav Chatterjee for pointing out the connection with [5]. We also thank anonymous referees for careful reading of the paper and many helpful suggestions which led to a much improved presentation.Attached Files
Submitted - 1205.2413v2.pdf
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Additional details
- Eprint ID
- 44291
- DOI
- 10.1016/j.spa.2013.10.007
- Resolver ID
- CaltechAUTHORS:20140313-090648465
- Created
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2014-03-13Created from EPrint's datestamp field
- Updated
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2021-11-10Created from EPrint's last_modified field