Markov processes with jump kernels decaying at the boundary
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916143735767040 |
|---|---|
| author | Cho, Soobin Kim, Panki Song, Renming Vondraček, Zoran |
| author_facet | Cho, Soobin Kim, Panki Song, Renming Vondraček, Zoran |
| contents | The goal of this work is to develop a general theory for non-local singular operators of the type $$
L^{\mathcal{B}}_αf(x)=\lim_{ε\to 0} \int_{D,\, |y-x|>ε}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-α}\,dy, $$ and $$ L f(x)=L^{\mathcal{B}}_αf(x) - κ(x) f(x), $$ in case $D$ is a $C^{1,1}$ open set in $\mathbb{R}^d$, $d\ge 2$. The function $\mathcal{B}(x,y)$ above may vanish at the boundary of $D$, and the killing potential $κ$ may be subcritical or critical.
From a probabilistic point of view we study the reflected process on the closure $\overline{D}$ with infinitesimal generator $L^{\mathcal{B}}_α$, and its part process on $D$ obtained by either killing at the boundary $\partial D$, or by killing via the killing potential $κ(x)$. The general theory developed in this work (i) contains subordinate killed stable processes in $C^{1,1}$ open sets as a special case, (ii) covers the case when $\mathcal{B}(x,y)$ is bounded between two positive constants and is well approximated by certain Hölder continuous functions, and (iii) extends the main results known for the half-space in $\mathbb{R}^d$. The main results of the work are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. Our results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. Our Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded $C^{1, 1}$ open sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Markov processes with jump kernels decaying at the boundary Cho, Soobin Kim, Panki Song, Renming Vondraček, Zoran Probability Analysis of PDEs 60Jxx, 60J45, 31C25, 35J08, 47G20, 60J46, 60J50, 60J76 The goal of this work is to develop a general theory for non-local singular operators of the type $$ L^{\mathcal{B}}_αf(x)=\lim_{ε\to 0} \int_{D,\, |y-x|>ε}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-α}\,dy, $$ and $$ L f(x)=L^{\mathcal{B}}_αf(x) - κ(x) f(x), $$ in case $D$ is a $C^{1,1}$ open set in $\mathbb{R}^d$, $d\ge 2$. The function $\mathcal{B}(x,y)$ above may vanish at the boundary of $D$, and the killing potential $κ$ may be subcritical or critical. From a probabilistic point of view we study the reflected process on the closure $\overline{D}$ with infinitesimal generator $L^{\mathcal{B}}_α$, and its part process on $D$ obtained by either killing at the boundary $\partial D$, or by killing via the killing potential $κ(x)$. The general theory developed in this work (i) contains subordinate killed stable processes in $C^{1,1}$ open sets as a special case, (ii) covers the case when $\mathcal{B}(x,y)$ is bounded between two positive constants and is well approximated by certain Hölder continuous functions, and (iii) extends the main results known for the half-space in $\mathbb{R}^d$. The main results of the work are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. Our results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. Our Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded $C^{1, 1}$ open sets. |
| title | Markov processes with jump kernels decaying at the boundary |
| topic | Probability Analysis of PDEs 60Jxx, 60J45, 31C25, 35J08, 47G20, 60J46, 60J50, 60J76 |
| url | https://arxiv.org/abs/2403.00480 |