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We establish subgeometric bounds on convergence rate of general Markovprocesses in the Wasserstein metric. In the discrete time setting we prove thatthe Lyapunov drift condition and the existence of a "good" $d$-small set implysubgeometric convergence to the invariant measure. In the continuous timesetting we obtain the same convergence rate provided that there exists a "good"$d$-small set and the Douc-Fort-Guillin supermartingale condition holds. As anapplication of our results, we prove that the Veretennikov-Khasminskiicondition is sufficient for subexponential convergence of strong solutions ofstochastic delay differential equations.

Subgeometric rates of convergence of Markov processes in the Wasserstein metric