On the non-existence of 2D, locally optimal ergodic and stationary matchings
Date:
I gave a short talk at Brown’s Graduate Lecture Series in Analysis and PDEs (GLESPA) on the work I conducted when writing my MSc thesis, which is available here.
Abstract
We disprove the existence of a locally optimal, stationary and ergodic matching between two Poisson Point Processes in $\mathbb{R}^2$, which extends the literature on matchings between points sampled uniformly at random from a bounded domain to the plane, and connects this highly-geometric problem with (discrete) optimal transport.
The main tool used in this analysis is the so-called Harmonic approximation theorem introduced by Goldman, Huessmann and Otto, which quantitavely analyses the well-known fact that the Monge-Ampere equation near the Lebesgue measure linearises to be the Poisson equation. All work is based on the following paper.