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The fixed points of branching Brownian motion

Abstract

In this work, we characterize all the point processes \(\theta =\sum _{i\in {\mathbb {N}}} \delta _{x_i}\) on \({\mathbb {R}}\) which are left invariant under branching Brownian motions with critical drift \(-\sqrt{2}\). Our characterization holds under the only assumption that \(\theta \) is locally finite and \(\theta ({\mathbb {R}}_+)<\infty \) almost surely.

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Data Availability Statement

All data generated or analyzed during this study are included in this article.

Notes

  1. For simplicity, we shall only consider the case of binary branching in this paper, but the main results hold also under the more general setting of [4].

  2. One possible way of verifying this is by writing the Laplace transform \({\mathbb {E}}[e^{-\langle {f,\theta _t}\rangle }]\) in terms of the solution of FKPP equation, see (3.1). Then using the fact that the limit in (2.16) exists, one can conclude that the Laplace transform converges to 1 if \(\theta _0([-x,-x+1]) = o(xe^{\sqrt{2}x})\) as \(x\rightarrow \infty \).

  3. A sharp asymptotics of \({\mathbb {P}}(M_t\ge \sqrt{2}t + x)\) is known for \(x= a \sqrt{t}\) or \(x=a t\), see [4, Lemma 4.5] and [14, Lemma 9.8]. However, since we require an estimate for all \(x\ge \sqrt{t}/\delta \), these estimates are not useful.

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Acknowledgements

We wish to thank Pascal Maillard and Bastien Mallein for useful discussions and Elie Aïdékon for pointing to us the reference [25] for the non-critical case. We also thank anonymous referees for carefully reading this article and providing their helpful comments. The research of X.C. is supported by ANR/FNS MALIN (Grant No. ANR-16-CE93-0003). The research of C.G. and A.S. is supported by the ERC grant LiKo 676999.

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Correspondence to Xinxin Chen.

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Dedicated to the memory of Tom Liggett.

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Chen, X., Garban, C. & Shekhar, A. The fixed points of branching Brownian motion. Probab. Theory Relat. Fields 185, 839–884 (2023). https://doi.org/10.1007/s00440-022-01183-4

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Keywords

  • Branching Brownian motion
  • Infinite particle system
  • Invariant measures
  • Decorated Poisson point process

Mathematics Subject Classification

  • 60G55
  • 60J80