论文标题
在对称$α$稳定过程中可接受的单数漂移中
On admissible singular drifts of symmetric $α$-stable process
论文作者
论文摘要
我们考虑了SDE存在（独特的）弱解决方案的问题，该SDE描述了对称$α$稳定的过程，该过程具有本地无界的漂移$ B：\ Mathbb r^d \ rightarrow \ Mathbb r^d $，$ d \ geq 3 $，$ 1 <α<2 $。在本文中，$ b $属于弱形式结合的矢量字段的类。后者出现是作为提供SDE背后的$ L^2 $理论的班级。 $ l^{\ frac {d} {α-1}} $ class和campanato-morrey class（因此，$ b $可能是如此单数，以至于它根据分数laplacian的热核会破坏标准的热内核估计）。我们表明，对于此类$ b $，操作员$ - （ - δ）^{\fracα{2}} - b \ cdot \ nabla $允许意识到是一个坠机发电机的实现，而fly feller semogroup（独特意义上）确定的概率措施是对相应的SDE的弱解决方案。该证明基于$（δ）^{\fracα{2}}} + b \ cdot \ nabla $ in $ l^p $，$ p> d-d-α + 1 $中的详细规则理论。
We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric $α$-stable process with a locally unbounded drift $b:\mathbb R^d \rightarrow \mathbb R^d$, $d \geq 3$, $1<α<2$. In this paper, $b$ belongs to the class of weakly form-bounded vector fields. The latter arises as the class providing the $L^2$ theory of the non-local operator behind the SDE, i.e.\,$(-Δ)^{\fracα{2}} + b \cdot \nabla$, and contains as proper sub-classes the other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, weak $L^{\frac{d}{α-1}}$ class and the Campanato-Morrey class (thus, $b$ can be so singular that it destroys the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that for such $b$ the operator $-(-Δ)^{\fracα{2}} - b \cdot \nabla$ admits a realization as a Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of $(-Δ)^{\fracα{2}} + b \cdot \nabla$ in $L^p$, $p>d-α+1$.
