Brownian motions on metric graphs

Werner, Florian

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URL:	https://madoc.bib.uni-mannheim.de/41517
URN:	urn:nbn:de:bsz:180-madoc-415178
Dokumenttyp:	Dissertation
Erscheinungsjahr:	2016
Ort der Veröffentlichung:	Mannheim
Hochschule:	Universität Mannheim
Gutachter:	Potthoff, Jürgen
Datum der mündl. Prüfung:	24 November 2016
Sprache der Veröffentlichung:	Englisch
Einrichtung:	Fakultät für Wirtschaftsinformatik und Wirtschaftsmathematik > Mathematik V (Potthoff -2020)
Fachgebiet:	510 Mathematik
Fachklassifikation:	MSC: 60J65, 60J40, 60J25, 47D07, 34B45, 05C99,
Normierte Schlagwörter (SWD):	Brownsche Bewegung , Metrischer Graph , Markov-Prozess
Freie Schlagwörter (Englisch):	Brownian motion , metric graphs , Markov processes , right processes , Feller–Wentzell boundary conditions
Abstract:	In this work, Brownian motions on metric graphs are defined as right continuous, strong Markov processes which, while inside an edge, are equivalent to the one-dimensional Brownian motion. Their generators are identified as Laplace operatorson the graph subject to non-local Feller–Wentzell boundary conditions at the vertices. Conversely, a pathwise construction is achieved for any set of admissible boundary conditions. This thesis generalizes the recent works of Kostrykin, Potthoff and Schrader, who examined Brownian motions on metric graphs which are continuous up to their lifetime. The theory is significantly complicated by the extension to the discontinuous setting. Here, the processes in question might feature jumps of infinite activity in the vicinity of any vertex, and their excursions from a vertex are not limited to adjacent edges. To overcome the challenges, transformation methods for Markov processes are surveyed and expanded in the modern context of Meyer–Getoor–Sharpe’s right processes. A universal revival method is established in order to concatenate various processes and to implement jump discontinuities. Probabilistic properties of Brownian motions on a metric graph are obtained, and their generators and resolvents are analyzed with the help of Dynkin’s formulas. By extending the results and the constructions of Itô–McKean’s fundamental paper on Brownian motions on the half line to the star graph case, the local description of all Brownian paths is achieved. By applying the transformation techniques and the Brownian properties, the local solutions are pasted together to obtain the process on the complete graph.