论文标题
与时间依赖的Kato类系数的强大存在和SDE解决方案的独特性
Strong existence and uniqueness of solutions of SDEs with time dependent Kato class coefficients
论文作者
论文摘要
考虑$ \ rd $中的随机微分方程（SDE）：$ dx_t = dw_t+b（t，x_t）\ d t $，其中$ w $是brownian Motion，$ b（\ cdot，\ cdot）$是一个可测量的矢量场。众所周知，如果$ | b |^2（\ cdot，\ cdot）= | b |^2（\ cdot）$属于kato类$ \ k_ {d，2} $，那么SDE的解决方案较弱。在本文中，我们表明，如果$ | b |^2 $属于Kato类$ \ k_ {d，\ a} $，对于某些$ \ a \ in（0,2）$（0,2）$（$ \ a $）可以任意接近$ 2 $），那么在现有的文献中，在现有文献中扩展了一个独特的强大解决方案，以扩展了一个独特的强大解决方案。此外，我们允许漂移与时间有关。我们为带有Kato类系数的抛物线方程解决方案的解决方案建立的新规律性估计起着至关重要的作用。
Consider stochastic differential equations (SDEs) in $\Rd$: $dX_t=dW_t+b(t,X_t)\d t$, where $W$ is a Brownian motion, $b(\cdot, \cdot)$ is a measurable vector field. It is known that if $|b|^2(\cdot, \cdot)=|b|^2(\cdot)$ belongs to the Kato class $\K_{d,2}$, then there is a weak solution to the SDE. In this article we show that if $|b|^2$ belongs to the Kato class $\K_{d,\a}$ for some $\a \in (0,2)$ ($\a$ can be arbitrarily close to $2$), then there exists a unique strong solution to the stochastic differential equations, extending the results in the existing literature as demonstrated by examples. Furthermore, we allow the drift to be time-dependent. The new regularity estimates we established for the solutions of parabolic equations with Kato class coefficients play a crucial role.
