Brownian Motions on Metric Graphs
  Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions. 
Vadim Kostrykin and J\"urgen Potthoff and Robert Schrader
math.PR