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<p>In this work, we resolve three open questions which were raised recently concerning the limiting extremal and cluster point processes of branching Brownian Motion. The former process records the heights of all extreme values of the motion, while the latter records the relative heights of extreme values in a genealogical neighborhood of order unity around a local maximum thereof. For the extremal point process, we show that first order asymptotics for its mass away from zero holds almost-surely, not just in-probability as was previously shown by Mytnik, L., Roquejoffre, J.-M., Ryzhik, L.: Fisher-KPP equation with small data and the extremal process of branching Brownian motion. Adv. Math. <b>396</b>, 108106 (2022) and Cortines, A., Hartung, L., Louidor, O.: The structure of extreme level sets in branching brownian motion. Ann. Probab. <b>47</b>(4), 2257–2302 (2019). We also provide a probabilistic proof for the emergence of the Stable-1 Law for the random fluctuations of the mass in that limit, thereby demystifying the source of these fluctuations, which was not apparent in the previous analytic derivation of the same result, by Mytnik, L., Roquejoffre, J.-M., Ryzhik, L.: Fisher-KPP equation with small data and the extremal process of branching Brownian motion. Adv. Math. <b>396</b>, 108106 (2022). Lastly, we also derive the explicit limiting law of the random fluctuations of the mass of the cluster point process, making rigorous and precise the derivation in the physics literature by Mueller, A., Munier, S.: Particle-number distribution in large fluctuations at the tip of branching random walks. Phys. Rev. E <b>102</b>(2), 022104 (2020) and Le, A.D., Mueller, A.H., Munier, S.: Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion. Europhys. Lett. <b>140</b>(5), 51003 (2022).</p>

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On the fluctuations of the extremal and cluster level sets of branching brownian motion

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Lisa Hartung,
  • Oren Louidor,
  • Tianqi Wu

摘要

In this work, we resolve three open questions which were raised recently concerning the limiting extremal and cluster point processes of branching Brownian Motion. The former process records the heights of all extreme values of the motion, while the latter records the relative heights of extreme values in a genealogical neighborhood of order unity around a local maximum thereof. For the extremal point process, we show that first order asymptotics for its mass away from zero holds almost-surely, not just in-probability as was previously shown by Mytnik, L., Roquejoffre, J.-M., Ryzhik, L.: Fisher-KPP equation with small data and the extremal process of branching Brownian motion. Adv. Math. 396, 108106 (2022) and Cortines, A., Hartung, L., Louidor, O.: The structure of extreme level sets in branching brownian motion. Ann. Probab. 47(4), 2257–2302 (2019). We also provide a probabilistic proof for the emergence of the Stable-1 Law for the random fluctuations of the mass in that limit, thereby demystifying the source of these fluctuations, which was not apparent in the previous analytic derivation of the same result, by Mytnik, L., Roquejoffre, J.-M., Ryzhik, L.: Fisher-KPP equation with small data and the extremal process of branching Brownian motion. Adv. Math. 396, 108106 (2022). Lastly, we also derive the explicit limiting law of the random fluctuations of the mass of the cluster point process, making rigorous and precise the derivation in the physics literature by Mueller, A., Munier, S.: Particle-number distribution in large fluctuations at the tip of branching random walks. Phys. Rev. E 102(2), 022104 (2020) and Le, A.D., Mueller, A.H., Munier, S.: Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion. Europhys. Lett. 140(5), 51003 (2022).