膨胀球中的分支布朗运动及其在温和障碍问题中的应用
Branching Brownian motion in an expanding ball and application to the mild obstacle problem
AI总结：
本文研究温和 Poisson 障碍中 BBM 总质量的大偏差与强大数定律，并借助次扩散膨胀球内 BBM 质量的大偏差结果完成证明。
摘要：
我们首先研究温和 Poisson 障碍环境中的 d 维分支布朗运动（BBM），其中 R^d 中的随机陷阱场由 Poisson 点过程生成。该陷阱场由以 Poisson 点过程的原子为中心、半径固定的球组成。温和障碍规则是：当粒子位于陷阱内部时，它们以较低的速率分支，该速率允许为零；而当位于陷阱外部时，它们以正常速率分支。我们证明了温和障碍中 BBM 总质量的大偏差概率上界，随后结合 Borel–Cantelli 引理，用这些上界证明了相应的强大数定律。我们的结果是淬火（quenched）型的，也就是说，它们对 Poisson 点过程下的几乎所有环境都成立。我们的强大数定律改进了文献 [6] 中已有的相应弱大数定律。然后，我们研究处于次扩散膨胀球内部的 d 维 BBM；球的边界具有失活作用，即一旦 BBM 的粒子碰到运动边界，它就会立即失活，但在之后的某个时刻可以被重新激活，条件是在该较晚时刻其祖先谱系完全位于膨胀球内部。当时间趋于无穷时，我们得到了关于球内质量异常小这一概率的大偏差结果。事实证明，温和障碍问题证明中的一个关键组成部分，正是关于膨胀球内 BBM 质量的大偏差结果。
英文摘要：
We first study a $d$-dimensional branching Brownian motion (BBM) among mild Poissonian obstacles, where a random trap field in $\mathbb{R}^d$ is created via a Poisson point process. The trap field consists of balls of fixed radius centered at the atoms of the Poisson point process. The mild obstacle rule is that when particles are inside traps, they branch at a lower rate, which is allowed to be zero, whereas when outside traps they branch at the normal rate. We prove upper bounds on the large-deviation probabilities for the total mass of BBM among mild obstacles, which we then use along with the Borel-Cantelli lemma to prove the corresponding strong law of large numbers. Our results are quenched, that is, they hold in almost every environment with respect to the Poisson point process. Our strong law improves on the existing corresponding weak law in [6]. Then, we study a $d$-dimensional BBM inside subdiffusively expanding balls, where the boundary of the ball is deactivating in the sense that once a particle of the BBM hits the moving boundary, it is instantly deactivated but can be reactivated at a later time provided its ancestral line is fully inside the expanding ball at that later time. We obtain a large-deviation result as time tends to infinity on the probability that the mass inside the ball is aytpically small. An essential ingredient in the proofs of the mild obstacle problem turns out to be the large-deviation result on the mass of BBM inside expanding balls.
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