Dirichlet problem for diffusions with jumps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929672141406208 |
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| author | Chen, Zhen-Qing Peng, Jun |
| author_facet | Chen, Zhen-Qing Peng, Jun |
| contents | In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$
$$
{\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x)
+ \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb R}^d$ that is uniformly elliptic and bounded, $b$ is an ${\mathbb R}^d$-valued function so that $|b|^2$ is in some Kato class ${\mathbb K}_d$, for each $x\in {\mathbb R}^d$, $J(x, dy)$ is a finite measure on ${\mathbb R}^d$ so that $x\mapsto J(x, {\mathbb R}^d)$ is in the Kato class ${\mathbb K}_d$. We show there is a unique Feller process $X$ having strong Feller property associated with ${\cal L}$, which can be obtained from the diffusion process having generator $ {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) $ through redistribution. We further show that for any bounded connected open subset $D\subset{\mathbb R}^d$ that is regular with respect to the Laplace operator $Δ$ and for any bounded continuous function $φ$ on $D^c$, the Dirichlet problem ${\cal L} u=0$ in $D$ with $u=φ$ on $D^c$ has a unique bounded continuous weak solution on ${\mathbb R}^d$. This unique weak solution can be represented in terms of the Feller process associated with ${\cal L}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirichlet problem for diffusions with jumps Chen, Zhen-Qing Peng, Jun Probability Analysis of PDEs Primary: 60H30, 35S15, 60J45, Secondary: 60G53 In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $$ {\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb R}^d$ that is uniformly elliptic and bounded, $b$ is an ${\mathbb R}^d$-valued function so that $|b|^2$ is in some Kato class ${\mathbb K}_d$, for each $x\in {\mathbb R}^d$, $J(x, dy)$ is a finite measure on ${\mathbb R}^d$ so that $x\mapsto J(x, {\mathbb R}^d)$ is in the Kato class ${\mathbb K}_d$. We show there is a unique Feller process $X$ having strong Feller property associated with ${\cal L}$, which can be obtained from the diffusion process having generator $ {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) $ through redistribution. We further show that for any bounded connected open subset $D\subset{\mathbb R}^d$ that is regular with respect to the Laplace operator $Δ$ and for any bounded continuous function $φ$ on $D^c$, the Dirichlet problem ${\cal L} u=0$ in $D$ with $u=φ$ on $D^c$ has a unique bounded continuous weak solution on ${\mathbb R}^d$. This unique weak solution can be represented in terms of the Feller process associated with ${\cal L}$. |
| title | Dirichlet problem for diffusions with jumps |
| topic | Probability Analysis of PDEs Primary: 60H30, 35S15, 60J45, Secondary: 60G53 |
| url | https://arxiv.org/abs/2501.06747 |