- Titel
- Contributions to measure-valued diffusion processes arising in statistical mechanics and population genetics
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-806718
- Datum der Einreichung
- 31.01.2022
- Datum der Verteidigung
- 08.09.2022
- Abstract (EN)
- 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.
- Freie Schlagwörter (EN)
- measure-valued diffusion, Langevin dynamic, Wasserstein metric, Aitchison geometry
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- angenommene Version / Postprint / Autorenversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-806718
- Veröffentlichungsdatum Qucosa
- 19.09.2022
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0