# American Institute of Mathematical Sciences

April  2014, 7(2): 207-238. doi: 10.3934/dcdss.2014.7.207

## From particles scale to anomalous or classical convection-diffusion models with path integrals

 1 Univ. La Rochelle, MIA CNRS EA 3165, F-17000 La Rochelle, France 2 Univ. d'Avignon et des Pays de Vaucluse, UMR 1114 EMMAH, F-84018 Avignon Cedex, France

Received  April 2013 Revised  July 2013 Published  September 2013

The present paper is devoted to the rigorous upscaling of some particles displacement model with trapping events, to the continuum scale. It focuses especially on the transitions between sub-diffusive and diffusive models. The work gives emphasis to the following points: 1. The distribution of waiting times in passing to the continuum limit. The common idea is that the distributions with slowly decaying long tails produce anomalous diffusion while the classical diffusion model corresponds to distributions with short tails. This is shown to be not always true by introducing a simple model of geometrical heterogeneity leading to trapping events without characteristic time scale. 2. The extension of the Feynman-Kac theory to some non-Brownian setting. 3. Constructing a microscopic random walk model that, thought based on a MIM approach, gives both fMIM and FFPE at the mesoscopic limit.
Citation: Catherine Choquet, Marie-Christine Néel. From particles scale to anomalous or classical convection-diffusion models with path integrals. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 207-238. doi: 10.3934/dcdss.2014.7.207
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