On diffusion processes with drift in $L_d$

Abstract

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.

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