  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.
