These notes contain all the material accumulated over six years in Strasbourg to teach "Quantum Probability" to myself and to an audience of commutative probabi

Quantum probability is a subtle blend of quantum mechanics and classical probability theory. Its important ideas can be traced to the pioneering work of Richard

Kolmogorov's forward, basic results -- Non-elliptic regularity results -- Preliminary elliptic regularity results -- Nash theory -- Localization -- On a manifol

The Heisenberg group comes from quantum mechanics and is the simplest non-commutative Lie group. While it belongs to the class of simply connected nilpotent Lie