arXiv 2001.049502020-01math.PR

关于$L_{d}$中的漂移扩散过程

On diffusion processes with drift in $L_{d}$

N. V. Krylov

备注：: 31 pages, some corrections and better explanations, one corollary added

摘要：

本文研究了椭圆算子$L=a^{ij}D_{ij}+b^{i}D_{i}$对应的马氏拟扩散过程的性质，该过程作用于$\mathbb{R}^{d}$上的函数，具有可测系数，有界和一致椭圆$a$和$b\in L_{d}(\mathbb{R}^{d})$ .证明了它们都是强马氏的强Feller转移半群$T_{t}$，对于某些$d_{0}\in (d/2, d)$，$T_{t}$也是$L_{d_{0}}(\mathbb{R}^{d})$中的连续有界半群.我们发现，$T_{t}$，$t>0$，有一个核心$p_{t}(x,y)$，这是可求和的$y$的权力$d_{0}/(d_{0}-1)$。这导致抛物线Aleksandrov估计与电源的可求和$d_{0}$，而不是通常的$d+1$。对于概率解，与这样一个过程，问题$Lu=f$在有界域$D\subset\mathbb{R}^{d}$与边界条件$u=g$，其中$f\in L_{d_{0}}(D)$和$g$是有界的，我们证明了它是Hölder连续的。这个问题的抛物线版本也被处理。我们还证明了Harnack不等式的调和和热量功能与这样的过程。最后，我们证明了概率解是$L_{d_{0}}$ -粘性解。

英文摘要：

We investigate properties of Markov quasi-diffusion processes corresponding to elliptic operators $L=a^{ij}D_{ij}+b^{i}D_{i}$, acting on functions on $\mathbb{R}^{d}$, with measurable coefficients, bounded and uniformly elliptic $a$ and $b\in L_{d}(\mathbb{R}^{d})$. We show that each of them is strong Markov with strong Feller transition semigroup $T_{t}$, which is also a continuous bounded semigroup in $L_{d_{0}}(\mathbb{R}^{d})$ for some $d_{0}\in (d/2, d)$. We show that $T_{t}$, $t>0$, has a kernel $p_{t}(x,y)$ which is summable in $y$ to the power of $d_{0}/(d_{0}-1)$. This leads to the parabolic Aleksandrov estimate with power of summability $d_{0}$ instead of the usual $d+1$. For the probabilistic solutions, associated with such a process, of the problem $Lu=f$ in a bounded domain $D\subset\mathbb{R}^{d}$ with boundary condition $u=g$, where $f\in L_{d_{0}}(D)$ and $g$ is bounded, we show that it is Hölder continuous. Parabolic version of this problem is treated as well. We also prove Harnack's inequality for harmonic and caloric functions associated with such a process. Finally, we show that the probabilistic solutions are $L_{d_{0}}$-viscosity solutions.

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摘要：

英文摘要：

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