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Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...

I learned from the gradient flow theory in Wasserstein space that an equation of gradient flow type $$\partial_t \rho + \nabla \cdot (\rho \nabla \frac{\delta F}{\delta \rho})=0,$$ can be derived as ...