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Contributions to measure-valued diffusion processes arising in statistical mechanics and population genetics

The present work is about measure-valued diffusion processes, which are
aligned with two distinct geometries on the set of probability measures.

In the first part we focus on a stochastic partial differential equation, the
Dean-Kawasaki equation, which can be considered as a natural candidate
for a Langevin equation on probability measures, when equipped with the
Wasserstein distance. Apart from that, the dynamic in question appears
frequently as a model for fluctuating density fields in non-equilibrium statistical
mechanics. Yet, we prove that the Dean-Kawasaki equation admits
a solution only in integer parameter regimes, in which case the solution is
given by a particle system of finite size with mean field interaction.

For the second part we restrict ourselves to positive probability measures on
a finite set, which we identify with the open standard unit simplex. We show
that Brownian motion on the simplex equipped with the Aitchison geometry,
can be interpreted as a replicator dynamic in a white noise fitness landscape.
We infer three approximation results for this Aitchison diffusion. Finally,
invoking Fokker-Planck equations and Wasserstein contraction estimates,
we study the long time behavior of the stochastic replicator equation, as an
example of a non-gradient drift diffusion on the Aitchison simplex.

Identiferoai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:80671
Date19 September 2022
CreatorsLehmann, Tobias
ContributorsUniversität Leipzig
Source SetsHochschulschriftenserver (HSSS) der SLUB Dresden
LanguageEnglish
Detected LanguageEnglish
Typeinfo:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text
Rightsinfo:eu-repo/semantics/openAccess

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