We briefly review Hida's white noise space theory and some  of its applications to linear stochastic systems and stochastic processes. To define derivatives and stochastic integrals we introduce a family of algebras of stochastic distributions, which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra. Non-commutative counterparts and applications to free stochastic processes and their derivatives will be also discussed. 

The talk is based on various collaborations, with in particular Haim Attia, Palle Jorgensen,  Alon Kipnis, David  Levanony, Guy Salomon and Tryphon Georgiou