Let's say P and Q be two different dirac delta probability measure, and suppose that $K_\sigma$ is a gaussian kernel. Let D be the wasserstein-2 distance.
It is known that $D(P,Q)=D(K_\sigma *P, K_\sigma*Q)$ by using the wasserstein-2 distance of gaussian. I've also seen a paper that relates heat kernel with wasserstein-2 distance.
Is it also true for any $P(x)=\frac{1}{n} \sum_{i=1}^n \delta(x-x_i)$ and $Q(x)=\frac{1}{n} \sum_{i=1}^n \delta(x-y_i)$? I don't know how to expand the previous result.
Is it also true for any $P$ and $Q$?
It is on $R^d$. Can we change all our above arguments to a compact space $\Omega$, Riemannian metric, and a corresponding heat kernel $H_\sigma$?
Any partial answer will be very thankful to me.