Harnack inequality for nonlocal operators

Abstract

Harnack inequalities are a fundamental property in both probability theory and analysis.The scale-variant Harnack inequalities play an important role in studying various properties, such as the regularity of harmonic functions in probability and analysis. This thesis focuses on scale-invariant Harnack inequalities for a class of nonlocal operators and a class of weakly coupled nonlocal operators. In Chapter 1, we show the scale-invariant elliptic Harnack inequality holds for a class of nonlocal operators, which are second order elliptic differential operators perturbed by non- local operators. We assume the existence of a conservative Hunt process corresponding to each operators $L$ in that class, and establish the scale-invariant Harnack inequalities for nonnegative functions that are $L$-harmonic. This is achieved by using Krylov estimate approach, where the comparison constant depends solely on the parameters of the class of the operators. We also establish the Hölder regularity for bounded $L$-caloric functions. In addition, we demonstrate that the scale-invariant parabolic Harnack inequality holds for nonnegative $L$-caloric functions and establish Hölder regularity for bounded $L$-caloric functions. In Chapter 2, utilizing the result from Chapter 1, we consider a system $G$ of nonlocaloperators $\{L_i,i = 1,..m\}$, as discussed in Chapter 1, connected by an index switching process $\{Λi,i = 1,...,m\}$ determined by its switching rate matrix $Q$. Such a system of operator whose coupling terms do not involve the derivatives of the unknown functions is called a weakly coupled nonlocal system. Weakly coupled systems are widely investigated in the field of physics, finance and engineering, etc. Through a piecing-together procedure, there exists a Hunt process corresponding to the weakly coupled operator $G$ wthin a certain class. Using the two-sided scale-invariant Green function estimates, we prove the scale-invariant Harnack inequalities for the weakly coupled nonlocal operators $G$. Under the irreducibility assumption of the switching matrix, we further derive a full rank scale-invariant Harnack inequality for this class of weakly coupled operators. The Appendix A serves as a complimentary part to Chapter 1. One of the essentialintermediate components in Krylov’s estimate approach presented in Chapter 1 is a lower bound of the hitting probability estimate. This result, along with other related important theorems, all ultimately relies on the equivalence between the Martingale problems and SDE for this class of nonlocal operators. However, there is limited literature available on this topic in English. Therefore, for both learning purposes and the reader’s convenience, we summarize and provide the detail explanations of some existing results originally in French.

In addition, the proof of the support theorem for diffusion processes and the Krylov’s estimate for diffusion operators have also been rewritten and clarified in details.