## Abstract

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∈ 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.

Original language | English (US) |
---|---|

Pages (from-to) | 165-199 |

Number of pages | 35 |

Journal | Probability Theory and Related Fields |

Volume | 179 |

Issue number | 1-2 |

DOIs | |

State | Published - Feb 2021 |

### Bibliographical note

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

## Keywords

- Diffusion processes
- Itô equations
- Markov processes