On diffusion processes with drift in L_d

2020/01/14 by Krylov, N. V.
#60J35 #60J60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2001.04950

Abstract

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 L_d₀(ℝ^d) for some d₀∈ (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 d₀/(d₀-1). This leads to the parabolic Aleksandrov estimate with power of summability d₀ 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⊂ℝ^d with boundary condition u=g, where f∈ L_d₀(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_d₀-viscosity solutions.

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