Probability Theory and Related Fields

, Volume 172, Issue 3–4, pp 615–662 | Cite as

On Brownian motion, simple paths, and loops

  • Artem SapozhnikovEmail author
  • Daisuke Shiraishi


We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in \((1,\frac{5}{3}]\). We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.

Mathematics Subject Classification

60K40 60D05 60G50 60K35 



Enormous thanks go to Alain-Sol Sznitman for his helpful discussions, encouragements, and fruitful comments. We also thank Yinshan Chang and Wendelin Werner for useful discussions and suggestions, and Yinshan Chang for a careful reading of the paper. This project was carried out while the second author was enjoying the hospitality of the Forschungsinstitut für Mathematik of the ETH Zürich and the Max Planck Institute for Mathematics in the Sciences. He wishes to thank these institutions. The research of the second author has been supported by the Japan Society for the Promotion of Science (JSPS). Finally, the second author thanks Hidemi Aihara for all her understanding and support.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeipzigLeipzigGermany
  2. 2.Department of MathematicsKyoto UniversityKyotoJapan

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