### Abstract:

While the stability of time-homogeneous Markov chains have been extensively studied through the concept of mixing times, the stability of time-inhomogeneous Markov chains has not been studied as in depth. In this manuscript we will introduce special types of time-inhomogeneous Markov chains that are defined through an adiabatic transition. After doing this, we define the adiabatic and the stable adiabatic times as measures of stability these special time-inhomogeneous Markov chains. To construct an adiabatic transition one needs to make a transitioning convex combination of an initial and final probability transition matrix over the time interval [0, 1] for two time-homogeneous, discrete time, aperiodic and irreducible Markov chains. The adiabatic and stable adiabatic times depend on how this convex combinations transitions. In the most general setting, we suggested that as long as P : [0, 1] --> P[superscript ia][subscript n] is a Lipschitz continuous function with respect to the ‖ ·‖₁ matrix norm, then the adiabatic time is bounded above by a function of the mixing time of the final probability transition matrix [equation] For the stable adiabatic time, the most general result we achieved was for nonlinear adiabatic transitions P[subscript ø (t)] = (1-ø (t))P₀+ ø(t)P₁ where ø is a Lipschitz continuous functions that is piecewise defined over a finite partition of the interval [0, 1] so that on each subinterval ø is a bi-Lipschitz continuous function. In this setting we asymptotically bounded the stable adiabatic time by the largest mixing of P[subscript ø(t)] over all t∈[0, 1]. We found that [equation] We also have some additional results at bound the stable adiabatic time in this manuscript, but they are included to show the different attempts we took and highlight how important it is to pick the right variables to compare. We also provide examples to queueing and statistical mechanics.