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Author: Heida, Martin × End date: 2023 × Start date: 2020 × Has files: Yes × Subject: stochastic homogenization × WGL Institute: WIAS ×

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    On quenched homogenization of long-range random conductance models on stationary ergodic point processes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Heida, Martin
    We study the homogenization limit on bounded domains for the long-range random conductance model on stationary ergodic point processes on the integer grid. We assume that the conductance between neares neighbors in the point process are always positive and satisfy certain weight conditions. For our proof we use long-range two-scale convergence as well as methods from numerical analysis of finite volume methods.
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    A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Franchi, Bruno; Heida, Martin; Lorenzani, Silvia
    In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.