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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:80671 |
| Date | 19 September 2022 |
| Creators | Lehmann, Tobias |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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