I would review results about construction and ergodicity of stochastic dynamics (of jump type) on inductive limit C^* algebra which satisfies the detailed balance condition in a Hilbert space associated to a nontrivial state. Additionally I should say few words about the set-up for Hypercontractivity and Logarithmic Sobolev inequality in noncommutative L_p spaces (including corresponding quantum entropy and - using an idea of particles structure - providing a class examples states for infinite quantum spin system for which the Logarithmic Sobolev inequality can be shown).