On diffusion processes with drift in Ld

• 成果类型: Article
• 署名作者: Krylov, N. V.
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01007-3
• 发表日期: 2021
• 页码: 165-199

摘要

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 R-d, with measurable coefficients, bounded and uniformly elliptic a and b is an element of 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 d(0) is an element of (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 solution, associated with such a process, of the problem Lu = f in a bounded domain D subset of R-d with boundary condition u = g, where f is an element of L-d0 (D) and g is bounded, we show that it is Holder 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.