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Asymptotic properties of Brownian motion delayed by inverse subordinators


Authors: Marcin Magdziarz and René L. Schilling
Journal: Proc. Amer. Math. Soc. 143 (2015), 4485-4501
MSC (2010): Primary 60G17; Secondary 60G52
Published electronically: April 6, 2015
MathSciNet review: 3373947
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Abstract: We study the asymptotic behaviour of the time-changed stochastic process $ \vphantom {X}^f\!X(t)=B(\vphantom {S}^f\!S (t))$, where $ B$ is a standard one-dimensional Brownian motion and $ \vphantom {S}^f\!S$ is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing Lévy process with Laplace exponent $ f$. This type of processes plays an important role in statistical physics in the modeling of anomalous subdiffusive dynamics. The main result of the paper is the proof of the mixing property for the sequence of stationary increments of a subdiffusion process. We also investigate various martingale properties, derive a generalized Feynman-Kac formula, the laws of large numbers and of the iterated logarithm for $ \vphantom {X}^f\!X$.


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Marcin Magdziarz
Affiliation: Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
Email: marcin.magdziarz@pwr.wroc.pl

René L. Schilling
Affiliation: Technische Universität Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany
Email: rene.schilling@tu-dresden.de

DOI: https://doi.org/10.1090/proc/12588
Received by editor(s): November 23, 2013
Received by editor(s) in revised form: July 1, 2014
Published electronically: April 6, 2015
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society