Excursions of diffusion processes and continued fractions

Alain Comtet 1, 2, Yves Tourigny 3

Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques 47 (2011) 850-874

It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We examine the probabilistic significance of the expansion. To illustrate our results, we discuss some examples of diffusions in deterministic and in random environments.

  • 1. Unite mixte de service de l'institut Henri Poincaré (UMSIHP),
    CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. School of Mathematics,
    University of Bristol