Rigorous derivations of diffusion systems from moderately interacting particle models

Abstract

This thesis is concerned with the derivation of certain types of nonlinear partial differential equations from stochastic interacting particle systems. The underlying methods are within the framework of mean-field limits, a well-known mathematical concept which has become an emerging tool of inter disciplinary research due to the increasing number of applications in population dynamics, physics, neuroscience, deep learning and others.The basic idea of these types of particle limits is to show that even though the particles are interacting with each other - under certain conditions - in the large particle limit,the system can be approximated by a density function which solves a partial differential equation: This is also called `propagation of chaos'. Throughout this thesis, the case of diffusive particle systems is considered leading to partial differential equations with positive diffusion parameters. Special focus in this work is put on moderately interacting particle systems, a technique where the interaction kernel of the particle system scales with the number of particles. In contrast to the classical mean-field limit, which is also called weak mean-field limit, the moderate regime leads to local partial differential equations.The thesis is split into three parts: In the first part, a rigorous derivation of a generalisedversion of the so-called SKT-system - a multi-species model from population dynamics -from moderately interacting particles is shown. In the second part, the method of moderately interacting particles is used to derive a porous media equation with fractional diffusion.Due to technical issues which occur in the moderate regime, rigorous estimates of non-local approximations of the particular partial differential equations are shown inthose two chapters, as well. The third part of this work shows a new technique for proving a conditional quantitative L2-convergence result for diffusive particles under the effectof aggregation, which can be seen as a step towards the proof of uctuations around theme an-field limit in the setting of aggregating particles.