Abstract:
Given a random sample $M_n= \{ x_1, ..., x_n\}$ uniformly distributed on a manifold $M$, we construct a weighted graph on $M_n$ by giving high weights to pairs of points that are at a distance smaller than a parameter epsilon. On the resulting graph we consider evolution equations that can be obtained as gradient flows of a free energy with respect to a discrete Wasserstein distance on the graph. We study the limit, as $n$ goes to infinity, of these discrete flows and provide conditions on the parameter epsilon that guarantee that, with probability one, they converge to an evolution equation on $M$ that can be characterized as a gradient flow of a free energy with respect to the Wasserstein space on $M$. |