Brownian motions on metric graphs

Werner, Florian

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URN: urn:nbn:de:bsz:180-madoc-415178
Document Type: Doctoral dissertation
Year of publication: 2016
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Potthoff, Jürgen
Date of oral examination: 24 November 2016
Publication language: English
Institution: School of Business Informatics and Mathematics > Mathematik V (Potthoff)
Subject: 510 Mathematics
Classification: MSC: 60J65, 60J40, 60J25, 47D07, 34B45, 05C99,
Subject headings (SWD): Brownsche Bewegung , Metrischer Graph , Markov-Prozess
Keywords (English): 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.

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