Processus de diffusion gouverne par la forme de Dirichlet de l’operateur de Schroedinger

It is well known that Schrödinger operator is unitary equivalent to the Dirichlet operator of the ground state measure, whenever the infimum of its spectrum is actually an eigenvalue. Moreover, the Markov process governed by this Dirichlet operator plays a crucial role in the so-called Feynes-Nelson stochastic mechanics. Unfortunately, none of the various attempts to construct this process seems to be satisfactory. Indeed, either the dimension is restricted to one, or non-explosion is not proved, or the actual state space is not completely known, or the assumptions are too restrictive. The aim of the present note is to give a construction of the desired diffusion process that works in full generality, and to give some properties of the transition functions. We would like to put the emphasis on the fact that all the probabilistic techniques we use are elementary or standard. The only new ingredients are recently discovered regularity properties of Schrödinger operator.