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Martin boundary of unbounded sets for purely discontinuous Feller processes
Kim, Panki,Song, Renming,Vondrač,ek, Zoran De Gruyter 2016 Forum mathematicum Vol.28 No.6
<P><B>Abstract</B></P><P>In this paper, we study the Martin kernels of general open sets associated with inaccessible points for a large class of purely discontinuous Feller processes in metric measure spaces. Let<I>D</I>be an unbounded open set. Infinity is accessible from<I>D</I>if the expected exit time from<I>D</I>is infinite, and inaccessible otherwise. We prove that under suitable assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from<I>D</I>. Similar results are also proved for finite boundary points of<I>D</I>.</P>
HEAT KERNEL ESTIMATES FOR DIRICHLET FRACTIONAL LAPLACIAN WITH GRADIENT PERTURBATION
Chen, Peng,Song, Renming,Xie, Longjie,Xie, Yingchao Korean Mathematical Society 2019 대한수학회지 Vol.56 No.1
We give a direct proof of the sharp two-sided estimates, recently established in [4, 9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1,1}$ open sets by using Duhamel's formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require D to be $C^{1,{\theta}}$ for some ${\theta}{\in}({\alpha}/2,1]$.
Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation
Peng Chen,Renming Song,Longjie Xie,Yingchao Xie 대한수학회 2019 대한수학회지 Vol.56 No.1
We give a direct proof of the sharp two-sided estimates, recently established in \cite{C-K-S-1, P-R}, for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel's formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,\theta}$ for some $\theta\in (\alpha/2, 1]$.
Non-symmetric Diffusions with Measure-valued Drifts and Potentials
Panki Kim,Renming Song 한국산업응용수학회 2007 한국산업응용수학회 학술대회 논문집 Vol.3 No.2
Recently, we have been working on non-symmetric diffusions with measure-valued drifts and potentials. In this survey article, we give a brief survey on the main results on these processes from[1-8].
Heat Kernel Estimates for Dirichlet Fractional Laplacian
Panki Kim,Zhen-Qing,Renming Song 한국산업응용수학회 2009 한국산업응용수학회 학술대회 논문집 Vol.2009 No.5
We consider the fractional Laplacian -(-Δ)<SUP>α/2</SUP> on an open subset in R<SUP>d</SUP> with zero exterior condition.We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C<SUP>1,1</SUP> open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C<SUP>1,1</SUP> open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.