Exit and Occupation times for Brownian Motion on Graphs with General Drift and Diffusion Constant

O. Benichou ¹, J. Desbois ²

Journal of Physics A Mathematical and Theoretical 42 (2009) 015004

We consider a particle diffusing along the links of a general graph possessing some absorbing vertices. The particle, with a spatially-dependent diffusion constant D(x) is subjected to a drift U(x) that is defined in every point of each link. We establish the boundary conditions to be used at the vertices and we derive general expressions for the average time spent on a part of the graph before absorption and, also, for the Laplace transform of the joint law of the occupation times. Exit times distributions and splitting probabilities are also studied and several examples are discussed.

• 1. Laboratoire de Physique Théorique des Liquides (LPTL),
  CNRS : UMR7600 – Université Paris VI – Pierre et Marie Curie
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
  CNRS : UMR8626 – Université Paris XI – Paris Sud