Heat kernel estimates for nonlocal kinetic operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916506764312576 |
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| author | Hou, Haojie Zhang, Xicheng |
| author_facet | Hou, Haojie Zhang, Xicheng |
| contents | In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ Δ^{α/2}_v + v \cdot \nabla_x, \quad α\in (0, 2),\ (x,v)\in {\mathbb R}^{d}\times{\mathbb R}^d,$$ where $ Δ^{α/2}_v $ represents the fractional Laplacian acting on the velocity variable $v$. Additionally, we establish logarithmic gradient estimates with respect to both the spatial variable $x$ and the velocity variable $v$. In fact, the estimates are developed for more general non-symmetric stable-like operators, demonstrating explicit dependence on the lower and upper bounds of the kernel functions. These results, in particular, provide a solution to a fundamental problem in the study of \emph{nonlocal} kinetic operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heat kernel estimates for nonlocal kinetic operators Hou, Haojie Zhang, Xicheng Probability Analysis of PDEs 60H10, 35K08, 82C40 In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ Δ^{α/2}_v + v \cdot \nabla_x, \quad α\in (0, 2),\ (x,v)\in {\mathbb R}^{d}\times{\mathbb R}^d,$$ where $ Δ^{α/2}_v $ represents the fractional Laplacian acting on the velocity variable $v$. Additionally, we establish logarithmic gradient estimates with respect to both the spatial variable $x$ and the velocity variable $v$. In fact, the estimates are developed for more general non-symmetric stable-like operators, demonstrating explicit dependence on the lower and upper bounds of the kernel functions. These results, in particular, provide a solution to a fundamental problem in the study of \emph{nonlocal} kinetic operators. |
| title | Heat kernel estimates for nonlocal kinetic operators |
| topic | Probability Analysis of PDEs 60H10, 35K08, 82C40 |
| url | https://arxiv.org/abs/2410.18614 |