  Computational optimal transport (OT) has recently emerged as a powerful
framework with applications in various fields. In this paper we focus on a
relaxation of the original OT problem, the entropic OT problem, which allows to
implement efficient and practical algorithmic solutions, even in high
dimensional settings. This formulation, also known as the Schr\"odinger Bridge
problem, notably connects with Stochastic Optimal Control (SOC) and can be
solved with the popular Sinkhorn algorithm. In the case of discrete-state
spaces, this algorithm is known to have exponential convergence; however,
achieving a similar rate of convergence in a more general setting is still an
active area of research. In this work, we analyze the convergence of the
Sinkhorn algorithm for probability measures defined on the $d$-dimensional
torus $\mathbb{T}_L^d$, that admit densities with respect to the Haar measure
of $\mathbb{T}_L^d$. In particular, we prove pointwise exponential convergence
of Sinkhorn iterates and their gradient. Our proof relies on the connection
between these iterates and the evolution along the Hamilton-Jacobi-Bellman
equations of value functions obtained from SOC-problems. Our approach is novel
in that it is purely probabilistic and relies on coupling by reflection
techniques for controlled diffusions on the torus.
