Stochastic calculus and martingales on trees

Jean Picard (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

Considering trees as simple examples of singular metric spaces, we work out a stochastic calculus for tree-valued processes. We study successively continuous processes and processes with jumps, and define notions of semimartingales and martingales. We show that martingales of class (D) converge almost surely as time tends to infinity, and prove on some probability spaces the existence and uniqueness of a martingale of class (D) with a prescribed integrable limit; to this end, we use either a coupling method or an energy method. This problem is related with tree-valued harmonic maps and with the heat semigroup for tree-valued maps.

Mots clés: Semimartingales on trees; Martingales on trees; Martingales with jumps; Coupling of diffusions; Harmonic maps; Energy minimizing maps

Domaines: Probabilités [math.PR]

Jean Picard. Stochastic calculus and martingales on trees. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2005, 41 (4), pp.Pages 631-683. ⟨10.1016/j.anihpb.2004.03.002⟩. ⟨hal-00472223⟩