Non-Linear Extensions of Classical and Quantum Stochastic Calculus and Essentially Infinite Dimensional Analysis

It is likely (at least for its proponent) that quantum probability, or more generally algebraic probability shall play for probability a role analogous to that played by algebraic geometry for geometry: many will complain against a loss of immediate intuition, but this is compensated for by an increase in power, the latter being measured by the capacity of solving old problems, not only inside probability theory, or at least of bringing non-trivial contributions to their advancement. The present, reasonably satisfactory, balance between developement of new techniques and problems effectively solved by these new tools should be preserved in order to prevent implosion into a self-substaining circle of problems and the main route to achieve this goal is the same as for classical probability, namely to keep a strong contact with advanced mathematical developement on one side and with real statistical data, wherever they come from, on the other.