Abstract
We prove a new result relating solutions of the scaled fractional Allen–Cahn equation to motion by mean curvature flow, motivated by the motion of hybrid zones in populations that exhibit long range dispersal. Our proof is purely probabilistic and takes inspiration from Etheridge et al. [30] to describe solutions of the fractional Allen–Cahn equation in terms of ternary branching α-stable motions. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators.
Funding Statement
The third author was supported by the ANID/Doctorado en el extranjero doctoral scholarship, grant number 2018-72190055. The first author was supported by the Engineering and Physical Sciences Research Council [EP/R513131/1].
Acknowledgments
We would like to thank the anonymous referee for their thorough reading and helpful suggestions. We are also grateful to Nic Freeman, Christina Goldschmidt and Alex Watson for their careful feedback.
Citation
Kimberly Becker. Alison Etheridge. Ian Letter. "Branching stable processes and motion by mean curvature flow." Electron. J. Probab. 29 1 - 59, 2024. https://doi.org/10.1214/24-EJP1087
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