# Robust Hölder Estimates for Parabolic Nonlocal Operators

@article{Chaker2019RobustHE, title={Robust H{\"o}lder Estimates for Parabolic Nonlocal Operators}, author={Jamil Chaker and M. Kassmann and Marvin Weidner}, journal={arXiv: Analysis of PDEs}, year={2019} }

In this work we study parabolic equations determined by nonlocal operators in a general framework of bounded and measurable coefficients. Our emphasis is on the weak Harnack inequality and H\"older regularity estimates for solutions of such equations. We allow the underlying jump measures to be singular with a singularity that depends on the coordinate direction. This approach also allows to study several classes of non-singular jump measures that have not been dealt with so far. The main… Expand

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#### References

SHOWING 1-10 OF 20 REFERENCES

Regularity estimates for elliptic nonlocal operators

- Mathematics
- 2015

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… Expand

Local Regularity for Parabolic Nonlocal Operators

- Mathematics
- 2013

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… Expand

Nonlocal operators with singular anisotropic kernels

- Mathematics
- Communications in Partial Differential Equations
- 2019

Abstract 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… Expand

Regularity results for nonlocal parabolic equations

- Mathematics
- 2013

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… Expand

Regularity of Harmonic functions for a class of singular stable-like processes

- Mathematics
- 2009

AbstractWe consider the system of stochastic differential equations
$$ dX_t=A(X_{t-})\, dZ_t, $$where Zt1 , . . . , Zdt are independent one-dimensional symmetric stable processes of order α, and the… Expand

Existence of densities for stable-like driven SDEʼs with Hölder continuous coefficients

- Mathematics
- 2013

Consider a multidimensional stochastic differential equation driven by a stable-like Levy process. We prove that the law of the solution immediately has a density in some Besov space, under some… Expand

Systems of equations driven by stable processes

- Mathematics
- 2006

AbstractLet Zjt, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations
where the matrix… Expand

The martingale problem for a class of nonlocal operators of diagonal type

- Mathematics
- 2018

We consider systems of stochastic differential equations of the form \[ \d X_t^i = \sum_{j=1}^d A_{ij}(X_{t-}) \d Z_t^j\] for $i=1,\dots,d$ with continuous, bounded and non-degenerate coefficients.… Expand

A Class of Singular Symmetric Markov Processes

- Mathematics
- 2013

We consider a class of pure jump Markov processes in ${\mathbb R}^d$ whose jump kernels are comparable to that of a certain d-dimensional Lévy process. Upper and lower bounds for the transition… Expand

Heat kernel estimates for stable-like processes on d-sets

- Mathematics
- 2003

The notion of d-set arises in the theory of function spaces and in fractal geometry. Geometrically self-similar sets are typical examples of d-sets. In this paper stable-like processes on d-sets are… Expand