MPEJ Volume 1, No.2, 37pp Received: June 17, 1995, Accepted: July 10, 1995 Adam W. Majewski, Boguslaw Zegarlinski Quantum Stochastic Dynamics I ABSTRACT: In the context of non-commutative $\BL_p$ spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice.