论文标题
在扩散过程中，漂移以$ l_ {d} $
On diffusion processes with drift in $L_{d}$
论文作者
论文摘要
我们调查了马尔可夫准延伸过程的属性，与椭圆运算符相对应的$ l = a^{ij} d_ {ij}+b^{i} d_ {i} $，在$ \ mathbb {r^}^{d} $上作用于函数，并带有可测量的系数，并限制了$ b的$ b. l_ {d}（\ Mathbb {r}^{d}）$。我们表明，他们每个人都是强大的马尔可夫，具有强大的砍伐过渡半群$ t_ {t} $，这也是$ l_ {d_ {0}}中的连续有限的半群，（\ Mathbb {r Mathbb {r}^{d}）$，对于某些$ d_ {0} \ in（d/d/d/d/d/d/d）$。我们表明$ t_ {t} $，$ t> 0 $，具有一个内核$ p_ {t}（x，y）$，在$ y $中可以汇总到$ d_ {0}/（d_ {0} -1）$的功率。这导致抛物线Aleksandrov估计具有总和的功率$ d_ {0} $，而不是通常的$ d+1 $。对于与此类过程相关的概率解决方案，与边界条件$ u = g $相关的问题$ lu = f $与边界条件$ u = g $，其中$ f \ in l_ {d_ {d_ {0}}（d）}（d）$ g $ in Bounde contuction thit It thit It the hould conter，此问题的抛物线版也得到了处理。我们还证明了Harnack对与此类过程相关的谐波和热量功能的不平等。最后，我们表明概率解决方案是$ l_ {d_ {0}} $ - 粘度解决方案。
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.
