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 H\"older 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.

中文翻译：

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在 $$L_{d}$$ 中漂移的扩散过程

我们研究了对应于椭圆算子 $L=a^{ij}D_{ij}+b^{i}D_{i}$ 的马尔可夫准扩散过程的性质，作用于 $\mathbb{R}^{d 上的函数}$，具有可测量的系数，有界且均匀椭圆 $a$ 和 $b\in L_{d}(\mathbb{R}^{d})$。我们证明了它们中的每一个都是具有强 Feller 转移半群 $T_{t}$ 的强马尔可夫，它也是 $L_{d_{0}}(\mathbb{R}^{d})$ 中的一个连续有界半群一些 $d_{0}\in (d/2, d)$。我们证明 $T_{t}$, $t>0$, 有一个核 $p_{t}(x,y)$，它可以在 $y$ 中求和到 $d_{0}/(d_{ 0}-1)$。这导致抛物线 Aleksandrov 估计具有可和性 $d_{0}$ 而不是通常的 $d+1$。对于与此过程相关联的问题 $Lu=f$ 在边界条件 $u=g$ 的有界域 $D\subset\mathbb{R}^{d}$ 中的概率解，