In this paper we continue our work on adiabatic time of time-inhomogeneous Markov chains first introduced in Kovchegov (2010) and Bradford and Kovchegov (2011). Our study is an analog to the well-known Quantum Adiabatic (QA) theorem which characterizes the quantum adiabatic time for the evolution of a quantum system as a result of applying of a series of Hamilton operators, each is a linear combination of two given initial and final Hamilton operators, i.e. $\mathbf{ H}(s) = (1-s)\mathbf{ H_0} + s\mathbf{ H_1}$. Informally, the quantum adiabatic time of a quantum system specifies the speed at which the Hamiltonian operators changes so that the ground state of the system at any time $s$ will always remain $\epsilon$-close to that induced by the Hamilton operator $\mathbf{ H}(s)$ at time $s$. Analogously, we derive a sufficient condition for the stable adiabatic time of a time-inhomogeneous Markov evolution specified by applying a series of transition probability matrices, each is a linear combination of two given irreducible and aperiodic transition probability matrices, i.e., $\mathbf{ P_{ t}} = (1-t)\mathbf{ P_{ 0}} + t\mathbf{ P_{ 1}}$. In particular we show that the stable adiabatic time $t_{ sad}(\mathbf{ P_{ 0}}, \mathbf{ P_{ 1}}, \epsilon) = O (t_{ mix}^{ 4}(\epsilon \slash 2) \slash \epsilon^{ 3}), $ where $t_{ mix}$ denotes the maximum mixing time over all $\mathbf{ P_{ t}}$ for $0 \leq t \leq 1$.
Categories: math.PR