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            <title xml:lang="en">From particles scale to anomalous or classical convection-diffusion models with path integrals</title>
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                <forename type="first">Catherine</forename>
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                <forename type="first">Marie-Christine</forename>
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            <idno type="stamp" n="UNIV-AVIGNON">Université d'Avignon</idno>
            <idno type="stamp" n="PREMIA">Mathématiques, Image et Applications</idno>
            <idno type="stamp" n="MIA" corresp="UNIV-ROCHELLE">Mathématiques, Image et Applications</idno>
            <idno type="stamp" n="AGROPOLIS">Agropolis</idno>
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            <idno type="stamp" n="UNIV-ROCHELLE">Université de La Rochelle</idno>
            <idno type="stamp" n="INRAE">Institut National de Recherche en Agriculture, Alimentation et Environnement</idno>
            <idno type="stamp" n="EMMAH" corresp="UNIV-AVIGNON">Environnement Méditerranéen et Modélisation des Agro-Hydrosystèmes</idno>
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                <term xml:lang="en">fractal mobile-immobile model</term>
                <term xml:lang="en">fractional Fokker-Planck equa- tion</term>
                <term xml:lang="en">fractional Feynman-Kac equation</term>
                <term xml:lang="en">diffusion versus sub-diffusion</term>
                <term xml:lang="en">up- scaling</term>
                <term xml:lang="en">continuum limit</term>
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              <p>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. In fact, 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 settings without handling with double Laplace transforms. This yields to better numerical simulations. 3. Constructing a microscopic random walk model that, thought based on a MIM approach, gives both fMIM and FFPE at the mesoscopic limit.</p>
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