Scaling limits of external multi-particle DLA on the plane and the supercooled Stefan problem
Scaling limits of external multi-particle DLA on the plane and the supercooled Stefan problem
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平面上外部多粒子 DLA 的尺度限制和过冷 Stefan 问题
DOI：
10.1214/22-aihp1330
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Xiling Zhang
中科院分区：
文献类型：
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作者：
S. Nadtochiy;Mykhaylo Shkolnikov;Xiling Zhang
我们考虑Rosenstock和马夸特在平面上的外部多粒子扩散限制聚集过程。基于[11]、[10]在一维空间中的最新发现，很自然地推测，在这样的模型中，增长的聚集体的标度极限由热方程的单相过冷Stefan问题的合适的“概率“公式中的增长的固相给出。为了解决这个猜想，我们将[10]中的概率公式扩展到多个空间维度。然后，我们表明，在过冷Stefan问题的固相的增长率的特征方程是满足外部MDLA过程的标度极限的不等式，这可以是严格的一般。在证明的过程中，我们建立了两个额外的结果有趣的在自己的权利：（i）的“交叉性质”的平面布朗运动的稳定性和（ii）之间的严格联系的概率解决方案的过冷Stefan问题和它的经典和弱的解决方案。
We consider (a variant of) the external multi-particle diffusion-limited aggregation (MDLA) process of Rosenstock and Marquardt on the plane. Based on the recent findings of [11], [10] in one space dimension it is natural to conjecture that the scaling limit of the growing aggregate in such a model is given by the growing solid phase in a suitable"probabilistic"formulation of the single-phase supercooled Stefan problem for the heat equation. To address this conjecture, we extend the probabilistic formulation from [10] to multiple space dimensions. We then show that the equation that characterizes the growth rate of the solid phase in the supercooled Stefan problem is satisfied by the scaling limit of the external MDLA process with an inequality, which can be strict in general. In the course of the proof, we establish two additional results interesting in their own right: (i) the stability of a"crossing property"of planar Brownian motion and (ii) a rigorous connection between the probabilistic solutions to the supercooled Stefan problem and its classical and weak solutions.
DOI：
10.2140/pmp.2022.3.171
发表时间：
2019-02
期刊：
Probability and Mathematical Physics
影响因子：
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作者：
F. Delarue;S. Nadtochiy;Mykhaylo Shkolnikov
通讯作者：
F. Delarue;S. Nadtochiy;Mykhaylo Shkolnikov