Stable Adiabatic Times For A Continuous Evolution Of Markov Chains

TL;DR

This paper introduces the concept of stable adiabatic time for continuous evolutions of irreducible, aperiodic Markov chains and bounds it using the chain's mixing time and Lipschitz continuity.

Contribution

It defines stable adiabatic time for time-inhomogeneous Markov chains governed by continuous matrix evolutions and provides an upper bound relating it to mixing time and Lipschitz constant.

Findings

Bound on stable adiabatic time in terms of mixing time.

Extension of stability analysis to continuous matrix evolutions.

Quantitative relationship involving Lipschitz constant.

Abstract

This paper continues the discussion on the stability of time-inhomogeneous Markov chains. In particular, this paper defines a time-inhomogeneous, discrete-time Markov chain governed by a continuous evolution in the appropriate martrix space. This matrix space, $\mathcal{P}_{n}^{ia}$, is the space of all stochastic matrices that are irreducible and aperiodic. For this new type of evolution there is a definition of a specific type of stability called the stable adiabatic time. This measure is bounded by a function of the optimal mixing time over the evolution. Namely, for a time-inhomogeneous, discrete-time Markov chain governed by a continuous evolution through a function $\mathbf{P}: [0,1] \rightarrow \mathcal{P}_{n}^{ia}$ and $0 < \epsilon < \frac{1}{2 \sqrt{n}}$ $$t_{sad}(\mathbf{P}, \epsilon) \leq \frac{3n^{3 \slash 2} L t_{mix}^{2}(\mathbf{P}_{\infty}, \epsilon)}{(1-2\sqrt{n} \epsilon) \epsilon}$$ \noindent where $L$ is a Lipschitz constant related to the function $\mathbf{P}$.