Moritz Kassmann

Non-local Dirichlet forms and symmetric jump processes

We consider the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f, f) + f Rd f(x) 2 dx, and where the jump kernel J satisfies for 0 < α < β < 2, |x - y| < 1. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E,F). We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

Harnack inequalities for non-local operators of variable order

We consider harmonic functions with respect to the operator formula math Under suitable conditions on n(x,h) we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator L is allowed to be anisotropic and of variable order.

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.

Local Regularity for Parabolic Nonlocal Operators

Weak solutions to parabolic integro-differential operators of order α ∈ (α0, 2) are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to α↗2. In this sense, the presentation is an extension of Mosers result from [20].

Harnack Inequalities: An Introduction

The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete.

Jump processes, L -harmonic functions, continuity estimates and the Feller property

Given a family of Levy measures � = {�(x, � )}x∈Rd we study the regu- larity of harmonic functions and the Feller property of corresponding jump processes. We establish estimates for harmonic functions which ensure com- pactness in the space of continuous functions but are weaker than Holder estimates. This approach allows us to work under quite weak assumptions on the familiy �. Our assumptions imply cases where there is no uniform lower bound on the probability of hitting sets before leaving a ball as the radius of the ball tends to zero. Cases where the dependence of �(x,M), M ∈ B(R d \ {0}), on the state variable x is only measurable and bounded

ON WEIGHTED POINCARÉ INEQUALITIES

The aim of this note is to show that Poincare inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincare inequalities are considered, too. The proof is short and does not involve covering arguments.

On Regularity for Beurling–Deny Type Dirichlet Forms

Characteristic examples of Beurling–Deny type Dirichlet forms are considered. The forms are identified with bilinear forms of integro-differential operators that arise as generators of jump-diffusion processes. The aim of this article is to prove Harnack inequalities for these operators and consequently Hölder regularity of weak H1-solutions. Mosers iteration technique is used.

Boundary Regularity for Nonlinear Elliptic Systems: Applications to the Transmission Problem

The Dirichlet problem for nonlinear elliptic systems with piecewise continuous coefficients is studied. The domain can be a n-dimensional polyhedron, possibly non-convex and exhibiting slits. Regularity in Nikolskii spaces up to the transmission surface and the boundary is proved.

Existence of a Generalized Green Function for Integro-Differential Operators of Fractional Order

It is a great pleasure and honor for both authors of this paper to contribute to this volume. Throughout their education in analysis, especially in the field of partial differential equations, the authors were in close contact with the ideas and works of O. A. Ladyzhenskaya, in particular through the well-known monographs [1]–[3] written by O. A. Ladyzhenskaya and her former students. The first author spent the academic year 1993/1994 as a graduate student in St.-Petersburg and thereby became acquainted with the famous school of analysis headed by O. A. Ladyzhenskaya.