Classical and Quantum Markov Processes Associated with q -Bessel Operators

We study the fundamental properties of classical and quantum Markov processes generated by q-Bessel operators and their extension to the algebra of all bounded operators on the Hilbert space Lq,α2. In particular, we find a suitable generalized Gorini-Kossakowski-Sudarshan-Lindblad representation for the infinitesimal generator of q-Bessel operator and show that both the classical and quantum Markov processes are transient for α 0 and recurrent for α = 0. We also show that they do not admit invariant states and, moreover that the support projection of any initial state instantaneously fills the full space.