Long-time $L^p$ Wasserstein contraction for diffusion processes without global dissipativity

The fact that a Markov diffusion semi-group on $\mathbb R^d$ contracts the $L^p$ Wasserstein distance, which has been extensively used to establish uniform-in-time stability estimates (e.g. with respect to numerical discretization errors), is a well-studied question in the case where the distances are in fact deterministically contracted by the drift (global dissipativity condition) or in the case $p=1$ (with reflection couplings). This work focuses on the non-globally dissipative case with $p>1$. This situation was previously considered in \cite{MonmarcheBruit}, but only for elliptic processes, and with a restriction on the diffusivity coefficient (which had to be large enough). Here, we extend this analysis to non-elliptic processes and provide sharper conditions to get contractions along synchronous coupling, including negative results, lower bounds and a characterization (at least in dimension 1) in terms of the maximal eigenvalue of a Feynman-Kac operator.