On diffusion processes with drift in L_d

We investigate properties of Markov quasi-diffusion processes corresponding to elliptic operators L= a^ijD_ij+ bⁱD_i, acting on functions on R^d, with measurable coefficients, bounded and uniformly elliptic a and b∈ L_d(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 Ld0(Rd) for some d∈ (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/ (d- 1). This leads to the parabolic Aleksandrov estimate with power of summability d instead of the usual d+ 1. For the probabilistic solution, associated with such a process, of the problem Lu= f in a bounded domain D⊂ R^d with boundary condition u= g, where f∈Ld0(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 Ld0-viscosity solutions.