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We are interested in the rate of convergence of a subordinate Markov processto its invariant measure. Given a subordinator and the corresponding Bernsteinfunction (Laplace exponent) we characterize the convergence rate of thesubordinate Markov process; the key ingredients are the rate of convergence ofthe original process and the (inverse of the) Bernstein function. At atechnical level, the crucial point is to bound three types of moments(sub-exponential, algebraic and logarithmic) for subordinators as time $t$tends to infinity. At the end we discuss some concrete models and we show thatsubordination can dramatically change the speed of convergence to equilibrium.

Subgeometric rates of convergence for Markov processes under subordination