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Heat kernels of non-symmetric jump processes: beyond the stable case Song, Renming
Description
Let $J$ be the Lévy density of a symmetric Lévy process in $\bf{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator \[ {\cal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \bf{R}^d: |z|>\epsilon\}}(f(x+z)-f(z))\kappa(x,z)J(z)\, dz\, , \] where $\kappa(x,z)$ is a Borel measurable function on $\bf{R}^d\times \bf{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and $|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$. We construct the heat kernel $p^\kappa(t, x, y)$ of ${\cal L}^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$. This talk is based on a joint paper with Panki Kim and Zoran Vondracek.
Item Metadata
Title |
Heat kernels of non-symmetric jump processes: beyond the stable case
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-11-10T11:42
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Description |
Let $J$ be the Lévy density of a symmetric Lévy process in
$\bf{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity.
Consider the non-symmetric and non-local operator
\[
{\cal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \bf{R}^d: |z|>\epsilon\}}(f(x+z)-f(z))\kappa(x,z)J(z)\, dz\, ,
\]
where $\kappa(x,z)$ is a
Borel measurable function on $\bf{R}^d\times \bf{R}^d$ satisfying
$0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and
$|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$.
We construct the heat kernel $p^\kappa(t, x, y)$ of ${\cal L}^\kappa$, establish its upper
bound as well as its fractional
derivative and gradient estimates. Under an additional weak upper scaling condition at infinity,
we also establish a lower bound for the heat kernel $p^\kappa$.
This talk is based on a joint paper with Panki Kim and Zoran Vondracek.
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Extent |
22 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Illinois
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Series | |
Date Available |
2017-06-18
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0348338
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International