305 research outputs found
Intrinsic scaling properties for nonlocal operators II
We study integrodifferential operators and regularity estimates for solutions
to integrodifferential equations. Our emphasis is on kernels with a critically
low singularity which does not allow for standard scaling. For example, we
treat operators that have a logarithmic order of differentiability. For
corresponding equations we prove a growth lemma and derive a priori estimates.
We derive these estimates by classical methods developed for partial
differential operators. Since the integrodifferential operators under
consideration generate Markov jump processes, we are able to offer an
alternative approach using probabilistic techniques.Comment: The assumptions have slightly been weakened. The material of
arXiv:1310.5371 has been integrate
Nonlocal quadratic forms with visibility constraint
Given a subset of the Euclidean space, we study nonlocal quadratic forms
that take into account tuples if and only if the line
segment between and is contained in . We discuss regularity of the
corresponding Dirichlet form leading to the existence of a jump process with
visibility constraint. Our main aim is to investigate corresponding
Poincar\'{e} inequalities and their scaling properties. For dumbbell shaped
domains we show that the forms satisfy a Poincar\'{e} inequality with diffusive
scaling. This relates to the rate of convergence of eigenvalues in singularly
perturbed domains
Markov chain approximations for symmetric jump processes
Markov chain approximations of symmetric jump processes are investigated.
Tightness results and a central limit theorem are established. Moreover, given
the generator of a symmetric jump process with state space \mathbbm{R}^d the
approximating Markov chains are constructed explicitly. As a byproduct we
obtain a definition of the Sobolev space H^{\alpha/2}(\mathbbm{R}^d), , that is equivalent to the standard one.Comment: 36 page
On weighted Poincar\'e inequalities
The aim of this note is to show that Poincar\'e inequalities imply
corresponding weighted versions in a quite general setting. Fractional
Poincar\'e inequalities are considered, too. The proof is short and does not
involve covering arguments.Comment: 6 page
Nonlocal operators with singular anisotropic kernels
We study nonlocal operators acting on functions in the Euclidean space. The
operators under consideration generate anisotropic jump processes, e.g., a jump
process that behaves like a stable process in each direction but with a
different index of stability. Its generator is the sum of one-dimensional
fractional Laplace operators with different orders of differentiability. We
study such operators in the general framework of bounded measurable
coefficients. We prove a weak Harnack inequality and H\"older regularity
results for solutions to corresponding integro-differential equations.Comment: 1 figure, 27 page
Regularity estimates for elliptic nonlocal operators
We study weak solutions to nonlocal equations governed by integrodifferential
operators. Solutions are defined with the help of symmetric nonlocal bilinear
forms. Throughout this work, our main emphasis is on operators with general,
possibly singular, measurable kernels. We obtain regularity results which are
robust with respect to the differentiability order of the equation.
Furthermore, we provide a general tool for the derivation of H\"{o}lder
a-priori estimates from the weak Harnack inequality. This tool is applicable
for several local and nonlocal, linear and nonlinear problems on metric spaces.
Another aim of this work is to provide comparability results for nonlocal
quadratic forms.Comment: 43 pages, 1 figur
Analysis of jump processes with nondegenerate jumping kernels
We prove regularity estimates for functions which are harmonic with respect
to certain jump processes. The aim of this article is to extend the method of
Bass-Levin[BL02] and Bogdan-Sztonyk[BS05] to more general processes.
Furthermore, we establish a new version of the Harnack inequality that implies
regularity estimates for corresponding harmonic functions
Regularity results for nonlocal parabolic equations
We survey recent regularity results for parabolic equations involving
nonlocal operators like the fractional Laplacian. We extend the results of
Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal
operators with kernels not being absolutely continuous with respect to the
Lebesgue measure.Comment: fixed some typos from the previous versio
The Dirichlet problem for nonlocal operators
In this note we set up the elliptic and the parabolic Dirichlet problem for
linear nonlocal operators. As opposed to the classical case of second order
differential operators, here the "boundary data" are prescribed on the
complement of a given bounded set. We formulate the problem in the classical
framework of Hilbert spaces and prove unique solvability using standard
techniques like the Fredholm alternative
Quadratic forms and Sobolev spaces of fractional order
We study quadratic functionals on that generate seminorms
in the fractional Sobolev space for . The
functionals under consideration appear in the study of Markov jump processes
and, independently, in recent research on the Boltzmann equation. The
functional measures differentiability of a function in a similar way as the
seminorm of . The major difference is that differences are taken into account only if lies in some double cone with apex
at or vice versa. The configuration of double cones is allowed to be
inhomogeneous without any assumption on the spatial regularity. We prove that
the resulting seminorm is comparable to the standard one of
. The proof follows from a similar result on discrete
quadratic forms in , which is our second main result. We
establish a general scheme for discrete approximations of nonlocal quadratic
forms. Applications to Markov jump processes are discussed
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