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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 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 $\theta\in (\alpha/2, 1]$.
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 ${\theta}{\in}({\alpha}/2,1]$.
Yinghao Qu,Jianting Zhou,Renming Liu,Leng Liao,Qingyuan Zhao 대한토목학회 2021 KSCE JOURNAL OF CIVIL ENGINEERING Vol.25 No.3
For large section members such as cables, there are insufficient methods for circumferential scanning along the cable sections for the detection of corroded broken wires. In this study, a detection method for broken wire damage (BWD) along the circumferential direction of cable based on the principle of self-magnetic flux leakage (SMFL) was proposed. First, the theoretical calculation model and the finite element model of the circumferential magnetic field in the polar coordinate system were established. The analysis results showed that the position of the BWD on the section of cable could be effectively judged by the concave position of the curves of the SMFL in the polar coordinate system. Through the circumferential scanning at different positions, the length of the BWD could be judged by the coordinates of x when the value of index “C” (similarity to circle) approached 1 for the first time in the x - C curves. Finally, the detection experiment for the BWD of a parallel wire bundle was carried out. The experimental results were consistent with the analysis results for the theoretical model and the finite element, and the experimental results verified the correctness of the detection method.
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>