Transition density estimates for diagonal systems of SDEs driven by cylindrical $\alpha$-stable processes
Tadeusz Kulczycki,Michał Ryznar +1 more
TLDR
In this paper, the authors considered the system of stochastic differential equation (SDE) driven by a cylindrical stable process and showed sharp two-sided estimates of the transition density of the process.Abstract:
We consider the system of stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and H\"older continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove H\"older and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.read more
Citations
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Nonlocal operators with singular anisotropic kernels
Jamil Chaker,Moritz Kassmann +1 more
TL;DR: In this paper, nonlocal operators acting on functions in the Euclidean space were studied, and the operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable proces.
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Heat kernels of non-symmetric Lévy-type operators
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Heat kernel bounds for nonlocal operators with singular kernels
TL;DR: In this paper, the fundamental solution for an integro-differential operator of order α in (0, 2)-dimensional Markov processes was shown to be equivalent to the Dirichlet form of one-dimensional jump processes, where the jumping measure is singular with respect to the Lebesgue measure.
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Regularity of solutions to anisotropic nonlocal equations
TL;DR: In this article, the Holder regularity of bounded harmonic functions with respect to solutions to stochastic systems of order was proved for one-dimensional symmetric stable processes of order.
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Fundamental solution for super-critical non-symmetric Lévy-type operators
TL;DR: In this paper, the heat kernel was constructed for the case when the order of the operator is positive and smaller than 1 (but without excluding higher orders up to 2) under certain assumptions.
References
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Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
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Heat kernel estimates for jump processes of mixed types on metric measure spaces
Zhen-Qing Chen,Takashi Kumagai +1 more
TL;DR: In this paper, the authors investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces, and derive sharp two-sided heat kernel estimate for such processes.
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Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators
TL;DR: In this article, a continuous transition density of the semigroup generated by the Kato class was constructed, where the transition density is comparable with that of the fractional Laplacian.
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On the existence of smooth densities for jump processes
TL;DR: In this paper, the authors consider a Levy process and the solution of a stochastic differential equation driven by it and obtain sufficient conditions ensuring the existence of a smooth density for the solution, which is similar to those of the Malliavin calculus for continuous diffusions.
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