Hastings-Levitov平面生长的波动结果
Fluctuation results for Hastings-Levitov planar growth
摘要：
在小粒子极限下，研究了Hastings-Levitov团簇外区域的涨落。这些被证明是由一个连续的高斯过程$\mathcal{F}$在空间中的全纯函数的值$\{ |z|>1 \}$，其中我们提供了一个明确的建设。$\mathcal{F}$的边界值$\mathcal{W}$被示出为在单位圆$\mathbb{T}$上的分布空间上执行Ornstein-Uhlenbeck过程，其可以被描述为随机分数阶热方程\[ \frac{\partial}{\partial t} \mathcal{W} (t,\vartheta ) = - (-Δ)^{1/2} \mathcal{W} (t,\vartheta ) + \sqrt{2}\, ξ(t, \vartheta ) \,, \]的解，其中$Δ$表示作用于空间分量的拉普拉斯算子，并且$ξ(t,\vartheta )$是时空白噪声。因此，我们发现，当集群是左无限增长，边界过程$\mathcal{W}$收敛到一个对数相关的分数高斯场，这可以实现为$(-Δ)^{-1/4}W$，$W$复白噪声$\mathbb{T}$。
英文摘要：
We study the fluctuations of the outer domain of Hastings-Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process $\mathcal{F}$ taking values in the space of holomorphic functions on $\{ |z|>1 \}$, of which we provide an explicit construction. The boundary values $\mathcal{W}$ of $\mathcal{F}$ are shown to perform an Ornstein-Uhlenbeck process on the space of distributions on the unit circle $\mathbb{T}$, which can be described as the solution to the stochastic fractional heat equation \[ \frac{\partial}{\partial t} \mathcal{W} (t,\vartheta ) = - (-Δ)^{1/2} \mathcal{W} (t,\vartheta ) + \sqrt{2}\, ξ(t, \vartheta ) \,, \] where $Δ$ denotes the Laplace operator acting on the spatial component, and $ξ(t,\vartheta )$ is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process $\mathcal{W}$ converges to a log-correlated Fractional Gaussian Field, which can be realised as $(-Δ)^{-1/4}W$, for $W$ complex White Noise on $\mathbb{T}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
