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Title of Journal: Acta Appl Math

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Abbravation: Acta Applicandae Mathematicae

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Springer Netherlands

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DOI

10.1007/s10530-010-9722-y

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1572-9036

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Quantitative Convergence Towards a SelfSimilar Pr

Authors: Hugues Berry Thomas Lepoutre Álvaro Mateos González
Publish Date: 2016/03/15
Volume: 145, Issue: 1, Pages: 15-45
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Abstract

Continuoustime random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion ie subdiffusion Here we describe the subdiffusive continuoustime random walk using agestructured partial differential equations with age renewal upon each walker jump where the age of a walker is the time elapsed since its last jump In the spatiallyhomogeneous zerodimensional case we follow the evolution in time of the age distribution An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a selfsimilar profile which corresponds to convergence to a stationary profile for the rescaled variables An important difficulty arises from the fact that the equation in selfsimilar variables is not autonomous and we do not have a specific analytical solution Therefore in order to quantify the latter convergence we estimate attraction to a timedependent “pseudoequilibrium” which in turn converges to the stationary profile


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