论文标题
Fisher-kpp方程，带有少量数据和分支布朗运动的极端过程
Fisher-KPP equation with small data and the extremal process of branching Brownian motion
论文作者
论文摘要
我们考虑二进制分支布朗尼运动粒子的极限过程$ {\ MATHCAL X} $。我们表明，在衍生品Martingale $ z $的对数转移之后，粒子的“密度”重新缩放了，该粒子的“密度”是$ n+x $从接近$ {\ Mathcal x} $的尖端的位置上的，它可能会收敛于概率的概率，向指示的多个Expiential $ e^x $ as $ n \ as $ n \ as $ n \ th++\ fty $。我们还表明，密度的波动，在另一个缩放和额外的随机但明确的偏移之后，收敛到$ 1 $稳定的随机变量。我们的方法使用分析技术，并由分支布朗运动的特性与溶液向Fisher-KPP方程的bramson移位之间的联系，并在\ cite {bd1，bd2}中引发了一些特定的初始条件，并在本文中进一步开发。 $ {\ Mathcal X} $的限制定理的证明至关重要地依赖于Fisher-KPP方程的Bramson Shift行为的精细渐近学，从初始条件开始的“ size” $ 0 <\ VAREPSILON \ ll 1 $的初始条件开始\ varepsilon^{ - 1}）]^{ - 1-γ}} $，有一些$γ> 0 $。
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are at distance $n+x$ from a position close to the tip of ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a $1$-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in \cite{BD1,BD2} and further developed in the present paper. The proofs of the limit theorems for ${\mathcal X}$ rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of "size" $0<\varepsilon\ll 1$, up to terms of the order $[{(\log \varepsilon^{-1})]^{-1-γ}}$, with some $γ>0$.
