Quantitative merging for time-inhomogeneous Markov chains in non-decreasing environments via functional inequalitities

We study time-inhomogeneous Markov chains to obtain quantitative results on their asymptotic behavior. We use Poincar\'e, Nash, and logarithmic-Sobolev inequalities. We assume that our Markov chain admits a finite invariant measure at each time and that the sequence of these invariant measures is non-decreasing. We deduce quantitative bounds on the merging time of the distributions for the chain started at two arbitrary points and we illustrate these new results with examples.