On diffusion processes with drift in Ld

N. V. Krylov

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate properties of Markov quasi-diffusion processes corresponding to elliptic operators L= aijDij+ biDi, acting on functions on Rd, with measurable coefficients, bounded and uniformly elliptic a and b∈ Ld(Rd). We show that each of them is strong Markov with strong Feller transition semigroup Tt, which is also a continuous bounded semigroup in Ld0(Rd) for some d∈ (d/ 2 , d). We show that Tt, t> 0 , has a kernel pt(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⊂ Rd 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.

Original languageEnglish (US)
Pages (from-to)165-199
Number of pages35
JournalProbability Theory and Related Fields
Volume179
Issue number1-2
DOIs
StatePublished - Feb 2021

Bibliographical note

Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

Keywords

  • Diffusion processes
  • Itô equations
  • Markov processes

Fingerprint

Dive into the research topics of 'On diffusion processes with drift in L<sub>d</sub>'. Together they form a unique fingerprint.

Cite this