  We present a novel kernel over the space of probability measures based on the
dual formulation of optimal regularized transport. We propose an Hilbertian
embedding of the space of probabilities using their Sinkhorn potentials, which
are solutions of the dual entropic relaxed optimal transport between the
probabilities and a reference measure $\mathcal{U}$. We prove that this
construction enables to obtain a valid kernel, by using the Hilbert norms. We
prove that the kernel enjoys theoretical properties such as universality and
some invariances, while still being computationally feasible. Moreover we
provide theoretical guarantees on the behaviour of a Gaussian process based on
this kernel. The empirical performances are compared with other traditional
choices of kernels for processes indexed on distributions.
