Abstract
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.
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Acknowledgements
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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Sapozhnikov, A., Shiraishi, D. On Brownian motion, simple paths, and loops. Probab. Theory Relat. Fields 172, 615–662 (2018). https://doi.org/10.1007/s00440-017-0817-6
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DOI: https://doi.org/10.1007/s00440-017-0817-6