Brownian Motions on Metric Graphs: Feller Brownian Motions on Intervals Revisited
The construction of the paths of all possible Brownian motions (in the sense of Knight) on a half line or a finite interval is reviewed.
math.PR↗
arXiv subjects
Jurgen Potthoff
Publications and source records attributed to Jurgen Potthoff.
Contraction semigroups on metric graphs
The main objective of the present work is to study contraction semigroups generated by Laplace operators on metric graphs, which are not necessarily self-adjoint. We prove criteria for such semigroups to be continuity and positivity preserving. Also we provide a characterization of generators of Feller semigroups on metric graphs.
math.FA↗
Heat kernels on metric graphs and a trace formula
We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.
math-ph↗