Feller Property and Infinitesimal Generator of the Exploration Process

We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale. The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.