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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All data generated or analyzed during this study are included in this article.
Notes
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].
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 \(x\rightarrow \infty \).
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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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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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DOI: https://doi.org/10.1007/s00440-022-01183-4
Keywords
- Branching Brownian motion
- Infinite particle system
- Invariant measures
- Decorated Poisson point process
Mathematics Subject Classification
- 60G55
- 60J80