Abstract
We analyse the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is the unique solution (up to a multiplicative constant) of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closed-form expressions.
Acknowledgments
The authors are deeply indebted to Stas Shaposhnikov for his help, patience, detailed explanations of some parts of the theory of FPK equations and for suggesting some useful ideas for the proofs. We are very grateful to Paolo Pigato and Peter Friz for fruitful discussions. We also would like to express our deep gratitude to the referee for thoroughly reading the paper and for offering very valuable suggestions. OB has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 683164), from the DFG Research Unit FOR 2402, and is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy — The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689, sub-project EF1-22). YY acknowledges partial support from ERC through Consolidator Grant 683164 (PI: Peter Friz).
Citation
Oleg Butkovsky. Vlad Margarint. Yizheng Yuan. "Law of the SLE tip." Electron. J. Probab. 28 1 - 25, 2023. https://doi.org/10.1214/23-EJP1015
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