Long‐time existence of Brownian motion on configurations of two landmarks

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Long‐time existence of Brownian motion on configurations of two landmarks. / Habermann, Karen; Harms, Philipp; Sommer, Stefan.

In: Bulletin of the London Mathematical Society, Vol. 56, No. 5, 2024, p. 1658-1679.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Habermann, K, Harms, P & Sommer, S 2024, 'Long‐time existence of Brownian motion on configurations of two landmarks', Bulletin of the London Mathematical Society, vol. 56, no. 5, pp. 1658-1679. https://doi.org/10.1112/blms.13018

APA

Habermann, K., Harms, P., & Sommer, S. (2024). Long‐time existence of Brownian motion on configurations of two landmarks. Bulletin of the London Mathematical Society, 56(5), 1658-1679. https://doi.org/10.1112/blms.13018

Vancouver

Habermann K, Harms P, Sommer S. Long‐time existence of Brownian motion on configurations of two landmarks. Bulletin of the London Mathematical Society. 2024;56(5):1658-1679. https://doi.org/10.1112/blms.13018

Author

Habermann, Karen ; Harms, Philipp ; Sommer, Stefan. / Long‐time existence of Brownian motion on configurations of two landmarks. In: Bulletin of the London Mathematical Society. 2024 ; Vol. 56, No. 5. pp. 1658-1679.

Bibtex

@article{b1b0b4c7537e43de9b5b75206a634ec5,
title = "Long‐time existence of Brownian motion on configurations of two landmarks",
abstract = "We study Brownian motion on the space of distinct landmarks in , 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 = "Karen Habermann and Philipp Harms and Stefan Sommer",
year = "2024",
doi = "10.1112/blms.13018",
language = "English",
volume = "56",
pages = "1658--1679",
journal = "Bulletin of the London Mathematical Society",
issn = "0024-6093",
publisher = "Oxford University Press",
number = "5",

}

RIS

TY - JOUR

T1 - Long‐time existence of Brownian motion on configurations of two landmarks

AU - Habermann, Karen

AU - Harms, Philipp

AU - Sommer, Stefan

PY - 2024

Y1 - 2024

N2 - We study Brownian motion on the space of distinct landmarks in , 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.

AB - We study Brownian motion on the space of distinct landmarks in , 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.

U2 - 10.1112/blms.13018

DO - 10.1112/blms.13018

M3 - Journal article

VL - 56

SP - 1658

EP - 1679

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 5

ER -

ID: 384495030