Metric diffusion at time \(t\geq 0\) along a foliation \(\mathcal{F}\) on a compact Riemannian manifold \((M,g)\) is defined as the Wasserstein distance \(d_W\) of Dirac masses diffused along the leafs of foliation, that is
\[
D_t d(x,y) = d_W(D_t\delta_x,D_t\delta_y),
\]
where \(D_t\) denotes the foliated diffusion operator. In the talk we present the topology of the metric space \((M, D_t d)\) and study the convergence in Wasserstein-Hausdorff topology of \((M,D_t d)_{t\geq 0}\) while \(t\to\infty\). We demonstrate the necessary condition for such convergence. In addition, for foliation of dimension one, the sufficient condition will be presented.
|
