Asymptotic properties of Brownian motion delayed by inverse subordinators

We study the asymptotic behaviour of the time-changed stochastic process fX(t) = B(fS(t)), where B is a standard one-dimensional Brownian motion and fS is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing Lévy process with Laplace exponen...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2015-10, Vol.143 (10), p.4485-4501
Main Authors: MAGDZIARZ, MARCIN, SCHILLING, RENÉ L.
Format: Article
Language:English
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Summary:We study the asymptotic behaviour of the time-changed stochastic process fX(t) = B(fS(t)), where B is a standard one-dimensional Brownian motion and fS 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 fX.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/12588