Probabilit\'es et fluctuations quantiques (Probabilities and quantum fluctuations)

This note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman’s interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations, which was brought to the forefront in some fractal approaches. It results, as in Nelson’s stochastic mechanics, in stochastic differential equations which are deduced from infinitesimal random walks. An extended english abstract gives most of the details.