Sharp Asymptotics for Regularised Optimal Transport

We study the small-regularisation limit for a unified family of regularised optimal transport problems, including entropically regularised optimal transport and $L^p$-regularised optimal transport. Regularisation is central to the practical use of optimal transport, both for computational reasons and for its connections with machine learning, numerical stability, and sparse transport plans.

Our main contribution is the derivation of sharp first-order asymptotics as the regularisation parameter tends to zero. For entropic optimal transport, we identify the exact bias induced by the regularisation under weaker assumptions than those previously available, including cases with unbounded support. For $L^p$-regularised optimal transport, we compute the sharp limiting constants for all $p>1$, extending known results beyond the quadratic case $p=2$.

The proof develops a unified and modular approach based on local Gaussian and Barenblatt-type profiles, together with a quantisation-based completion of the marginal constraints. This framework separates the local computation of the sharp constant from the global construction of admissible couplings, and also clarifies the connection between regularised optimal transport and nonlinear diffusion limits.

This work is currently in preparation.