Energy methods for nonsymmetric nonlocal operators

Weidner M. Energy methods for nonsymmetric nonlocal operators. Bielefeld: Universität Bielefeld; 2022.The goal of this thesis is to develop the regularity theory for nonlocal parabolic equations. We focus on problems driven by nonlocal operators associated with nonsymmetric bilinear forms. In contrast to the symmetric case, nonsymmetric nonlocal operators have not yet been studied systematically. Our main results contain Hölder regularity estimates, full Harnack inequalities, and two-sided heat kernel estimates, which we obtain by extending recently established nonlocal energy methods. We are able to connect the theory of nonsymmetric nonlocal operators with the important results of Aronson-Serrin, Ladyzhenskaya-Solonnikov-Ural'tceva, and Aronson in the local linear case. Moreover, we develop a new technique for proving upper off-diagonal heat kernel estimates for nonlocal operators, which is based on the original ideas by Aronson in the local case. As an application of our regularity results, we establish Markov chain approximations for a large class of nonsymmetric diffusions and jump processes