论文标题
超棕色运动的绝对连续性具有无限平均值
Absolute continuity of the Super-Brownian motion with infinite mean
论文作者
论文摘要
在这项工作中，我们证明，对于任何维度$ d \ geq 1 $和（0,1）$ super-brownian运动的任何$γ\ in（0,1）$ super-brownian运动，与log-laplace方程相对应\ begin {equation*} \ begin {split} \ frac {\ frac {\ frac {\ partial v（t，x，x，x，x）} { v(t,x) + v^γ(t,x) ,\: (t,x) \in \mathbb{R}_+\times \mathbb{R}^d,\\ v(0,x)&= f(x) \end{split} \end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$.我们的证明基于\ ll \方程解决方案的属性。 我们还证明，当初始数据$ v（0，\ cdot）是一个有限的，非零的度量时，\ ll \ equation具有独特的连续解决方案。此外，该解决方案不断取决于初始数据。
In this work we prove that for any dimension $d\geq 1$ and any $γ\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \begin{equation*} \begin{split} \frac{\partial v(t,x)}{\partial t } & = \frac{1}{2}\bigtriangleup v(t,x) + v^γ(t,x) ,\: (t,x) \in \mathbb{R}_+\times \mathbb{R}^d,\\ v(0,x)&= f(x) \end{split} \end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \LL\ equation. We also prove that when initial datum $v(0,\cdot)$ is a finite, non-zero measure, then the \LL\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.
