| Speaker: Zoran Vondracek Affiliation: University of Zagreb Time: 2:00 pm Friday, 2 March, 2018 Location: MLT2/303-102 |
| The classical boundary Harnack principle (BHP) is a statement asserting that two positive harmonic functions vanishing on the portion of the boundary decay at the same rate. In the first part of the talk, I will give a brief overview of the BHP and the related Carleson estimate in the case of the classical Laplacian, or probabilistically, Brownian motion. Then I will discuss the BHP for certain non-local operators, or in the probabilistic language, for certain isotropic Lévy processes, a typical example being the isotropic stable process. Due to discontinuity of such processes (as opposed to Brownian motion) regularity of the boundary does not play a role and more refined version of the BHP can be established. In case of a smooth boundary, one can often obtain the exact decay rate of positive harmonic functions. Finally, in the last part of the talk, I will address some very recent results about BHP for subordinate killed Brownian motion in smooth domains (as well as for some related processes). The new feature is that for some of those processes the BHP can fail (although the Carleson estimate is valid). |