  This article is a mathematical analysis of the Open Quantum Brownian Motion.
This object was introduced by Bernard, Bauer, Benoist and Tilloy as the limit
of a family of Open Quantum Random Walks on the discrete line. We prove the
convergence for the three possible descriptions of this object: the quantum
trajectory satisfying a Belavkin Equation, the unitary evolution on the Fock
space satisfying a quantum Langevin Equation, and the Lindbladian evolution. We
introduce a very general framework for the continual measurement of
non-demolition observables, which is applied to the measurement of the position
of the Open Quantum Brownian Motion, and we probe some questions related to the
convergence of processes in this context.
