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PreprintCC BY 4.0arXiv

Stochastic Completeness for Landmark Shape Spaces with Induced Metrics

  • Karen Habermann
  • Mao Nishino
  • Stefan Sommer
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Abstract

Landmark shape spaces arise from configurations of n distinct landmarks with rigid motions factored out and scale normalized. When equipped with sufficiently regular metrics inherited from right-invariant metrics on diffeomorphism groups, the shape spaces inherit geodesic completeness from geodesics of diffeomorphisms. We now answer the corresponding stochastic question if the Riemannian Brownian motion on landmark shape spaces exists for all time and hence rule out initially distinct landmarks colliding when following a Brownian flow. We show that with general classes of metrics, including degenerate metrics with infinitesimal rigid motions in their null space, landmark shape spaces are stochastically complete, thus making the use of Riemannian Brownian motion for modelling shape stochasticity in applied fields well-founded.

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Cite

BibTeXRIS
BibTeX
@misc{habermann2026stochastic,
  title = {{Stochastic Completeness for Landmark Shape Spaces with Induced Metrics}},
  author = {Karen Habermann and Mao Nishino and Stefan Sommer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39845},
  eprint = {2609.39845},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39845},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stochastic Completeness for Landmark Shape Spaces with Induced Metrics
AU  - Karen Habermann
AU  - Mao Nishino
AU  - Stefan Sommer
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39845
UR  - https://arxiv.org/abs/2609.39845
AB  - Landmark shape spaces arise from configurations of n distinct landmarks with rigid motions factored out and scale normalized. When equipped with sufficiently regular metrics inherited from right-invariant metrics on diffeomorphism groups, the shape spaces inherit geodesic completeness from geodesics of diffeomorphisms. We now answer the corresponding stochastic question if the Riemannian Brownian motion on landmark shape spaces exists for all time and hence rule out initially distinct landmarks colliding when following a Brownian flow. We show that with general classes of metrics, including degenerate metrics with infinitesimal rigid motions in their null space, landmark shape spaces are stochastically complete, thus making the use of Riemannian Brownian motion for modelling shape stochasticity in applied fields well-founded.
ER  -