周期介质中分支扩散过程的渐近行为
Asymptotic behavior of branching diffusion processes in periodic media
摘要：
研究了周期介质中分支扩散过程的渐近行为。对于一个超临界分支过程，我们区分两种类型的行为的规范化数量的粒子在一个有界的域，取决于该域的距离，从该区域的大部分粒子所在。在随时间线性增长的距离处，我们观察到不连续性（即，对于某些$k$，$k$阶矩支配一阶矩的$k$阶次），而在随时间亚线性增长的距离处，我们表明所有矩都收敛。在我们的分析中的一个关键因素是一个尖锐的估计过渡内核的分支过程，有效的线性时间距离的初始粒子的位置。
英文摘要：
We study the asymptotic behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the $k$-th moment dominates the $k$-th power of the first moment for some $k$), while, at distances that grow sub-linearly in time, we show that all the moments converge. A key ingredient in our analysis is a sharp estimate of the transition kernel for the branching process, valid up to linear in time distances from the location of the initial particle.
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