论文标题
用局部界限系数的1D退化扩散方程的基本解决方案
Fundamental Solution to 1D Degenerate Diffusion Equation with Locally Bounded Coefficients
论文作者
论文摘要
在这项工作中，我们研究退化扩散方程$ \ partial_ {t} = x^αa\ left（x \ right）\ partial_ {x}^{2}^{2}+b \ left（x \ right）\ partial_ {x} $带有Cauchy初始数据和Dirichlet边界条件为$ 0 $。我们假设扩散运算符0处的退化顺序为$α\ in \ left（0,2 \右）$，$ a \ left（x \ right）$和$ b \ left（x \ right）$仅本地限制。我们采用了概率方法和分析方法的组合：通过分析潜在扩散过程的行为，我们为基本解决方案$ p \ left（x，y，t \ right）$提供了明确的结构，并证明了$ p \ left的几个属性（x，x，y，y，t \ right）$;通过执行本地化程序，我们获得了$ p \ left的近似值（x，y，t \ right）$，$ x，y $在0和$ t $的附近，足够小，其中错误估计仅依赖于$ a \ a \ a \ a \ aff（x \ right）$和$ b \ weft（x \ right）$（x \ right）$（和他们的derivivitations）的局部界限。在$α= 1 $的情况下，有关于这种退化扩散的丰富文献。我们的工作将现有结果的一部分扩展到了分析环境中（例如，对基本解决方案的热内核估计）和概率观点（例如，随机微分方程的良好性）。
In this work we study the degenerate diffusion equation $\partial_{t}=x^αa\left(x\right)\partial_{x}^{2}+b\left(x\right)\partial_{x}$ for $\left(x,t\right)\in\left(0,\infty\right)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $α\in\left(0,2\right)$, and both $a\left(x\right)$ and $b\left(x\right)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $p\left(x,y,t\right)$ and prove several properties for $p\left(x,y,t\right)$; by conducting a localization procedure, we obtain an approximation for $p\left(x,y,t\right)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $a\left(x\right)$ and $b\left(x\right)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $α=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations).
