Sticky Brownian motion is a stochastic process that alternates standard Brownian motion inside a given domain and tangential diffusion along its boundary. At the PDE level, the corresponding Fokker-Planck diffusion consists in a bulk/interface coupled system of parabolic PDEs with Wentzell boundary conditions. In a joint paper with Casteras and Santambrogio we proved that the system of PDEs can be characterized as a gradient flow in the space of probability measures by using smooth Wasserstein optimal transport. When the diffusion on the boundary becomes infinitely fast the geometry degenerates, and the domain collapses to a singular sphere with a pointy geometry/Laplacian and nonlocal boundary conditions. I will show how, by choosing the right geometric point of view, the variational structure survives and the analysis still carries through.

Joint work with Rui Silva.