Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise
Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise
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DOI：
10.1214/16-aihp796
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发表时间：
2014-12
期刊：
影响因子：
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通讯作者：
V. Knopova;A. Kulik
中科院分区：
文献类型：
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作者：
V. Knopova;A. Kulik
令$L：= -a（x）（-\Delta）^{\alpha/2}+（B（x），\nabla）$，其中$\alpha\in（0，2）$，$a：\rd\to（0，\infty）$，$B：\rd\to \rd$.在对系数a和B的正则性假设下，我们将（L，C_\infty^2（\rd））的C\infty（\rd）-闭包与一个具有转移概率密度p_t（x，y）的Feller Markov过程X联系起来.为了构造这个转移概率密度并得到它的双边估计，我们发展了一种新的参数化方法，它允许我们处理$0<\alpha\leq 1$和$B\neq 0$的情况，即当生成元的梯度部分不受跳跃部分支配时。
Let $L:= -a(x) (-\Delta)^{\alpha/2}+ (b(x), \nabla)$, where $\alpha\in (0,2)$, and $a:\rd\to (0,\infty)$, $b: \rd\to \rd$. Under certain regularity assumptions on the coefficients $a$ and $b$, we associate with the $C_\infty(\rd)$-closure of $(L, C_\infty^2(\rd))$ a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$. To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which allows us to handle the case $0<\alpha\leq 1$ and $b\neq 0$, i.e. when the gradient part of the generator is not dominated by the jump part..