Mathematics > Probability
[Submitted on 5 Jul 2018 (v1), last revised 12 Dec 2019 (this version, v2)]
Title:General self-similarity properties for Markov processes and exponential functionals of L{é}vy processes
View PDFAbstract:Positive self-similar Markov processes (pssMp) are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed Lévy process via Lamperti representation. In this work, we are interested in the following problem: what happens if we consider Markov processes in dimension $1$ or $2$ that satisfy self-similarity properties of a more general form than a scaling property ? Can they all be represented as a function of a time-changed Lévy process ? If not, how can Lamperti representation be generalized ? We show that, not surprisingly, a Markovian process in dimension $1$ that satisfies self-similarity properties of a general form can indeed be represented as a function of a time-changed Lévy process, which shows some kind of universality for the classical Lamperti representation in dimension $1$. However, and this is our main result, we show that a Markovian process in dimension $2$ that satisfies self-similarity properties of a general form is represented as a function of a time-changed exponential functional of a bivariate Lévy process, and processes that can be represented as a function of a time-changed Lévy process form a strict subclass. This shows that the classical Lamperti representation is not universal in dimension $2$. We briefly discuss the complications that occur in higher dimensions. In dimension $2$ we present an example, built from a self-similar fragmentation process, where our representation in term of an exponential functional of a bivariate Lévy process appears naturally and has a nice interpretation in term of the self-similar fragmentation process.
Submission history
From: Grégoire Véchambre [view email] [via CCSD proxy][v1] Thu, 5 Jul 2018 07:35:47 UTC (64 KB)
[v2] Thu, 12 Dec 2019 02:56:10 UTC (73 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.