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Long-time existence of Brownian motion on configurations of two landmarks by Habermann, Karen; Harms, Philipp; Sommer, Stefan is a scholarly article available to read on EtoBox.

What is Long-time existence of Brownian motion on configurations of two landmarks about?

We study Brownian motion on the space of distinct landmarks in $\mathbb{R}^d$, considered as a homogeneous space with a Riemannian metric inherited from a right-invariant metric on the diffeomorphism group. As of yet, there is no proof of long-time existence of this process, despite its fundamental importance in statistical shape analysis, where it is used to model stochastic shape evolutions. We make some first progress in this direction by providing a full classification of long-time existence for configurations of exactly two landmarks, governed by a radial kernel. For low-order Sobolev kernels, we show that the landmarks collide with positive probability in finite time, whilst for higher-order Sobolev and Gaussian kernels, the landmark Brownian motion exists for all times. We illustrate our theoretical results by numerical simulations.

Author

Habermann, Karen; Harms, Philipp; Sommer, Stefan

Published

2023

Language

EN