Fractional heat equation - Revision history

imported>Moritz at 21:55, 8 July 2012

2012-07-08T21:55:40Z

← Older revision		Revision as of 16:55, 8 July 2012
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	The fractional heat kernel $p(t,x)$ is the fundamental solution to the fractional heat equation. It is the function which solves the equation		The fractional heat kernel $p(t,x)$ is the fundamental solution to the fractional heat equation. It is the function which solves the equation
	\begin{align*}		\begin{align*}
	p(0,x) &= \~~delta_0~~ \\		p(0,x) &= \delta_{\{x\}} \\
	p_t(t,x) + (-\Delta)^s p &= 0		p_t(t,x) + (-\Delta)^s p &= 0
	\end{align*}		\end{align*}
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	\[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]		\[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]

	More generally, the heat kernel can be shown to exists for nonlocal regular Dirichlet forms $(\mathcal{E}, D(\mathcal{E}))$. Assume		More generally, the heat kernel can be shown to exists for certain nonlocal regular Dirichlet forms $(\mathcal{E}, D(\mathcal{E}))$. Assume
	\[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \]		\[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \]
	and $D(\mathcal{E})$ is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \\|u\\|^2_{L^2}$.		and $D(\mathcal{E})$ is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \\|u\\|^2_{L^2}$.
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	Then the corresponding transition semigroup has a heat kernel $p(t,x,y)$ under quite general assumptions on $J(x,y)$<ref name="BBCK09"/>.		Then the corresponding transition semigroup has a heat kernel $p(t,x,y)$ under quite general assumptions on $J(x,y)$<ref name="BBCK09"/>.

	If $J(x,y)$ is comparable to $\|x-y\|^{-d-\alpha}~~$ for small values of $\|x-y\|$ and under some mild assumptions on $J(x,y)$ for large values of $\|x-y\|~~$, $p(t,x,y)$ satisfies a bound like above <ref name="BL02"/>. One can relax the assumptions significantly and still prove sharp bounds for small time as well as for large time. <ref name="CKK11"/>.		If $J(x,y)$ is comparable to $\|x-y\|^{-d-\alpha}$, $p(t,x,y)$ satisfies a bound like above <ref name="BL02"/><ref name="ChKu03"/>. One can relax the assumptions significantly and still prove sharp bounds for small time as well as for large time <ref name="CKK11"/>.


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	{{reflist\|refs=		{{reflist\|refs=
	<ref name="CKK11">		<ref name="CKK11">
	{{Citation \| ~~last1~~=Kumagai \| ~~first1~~=Takashi \| last2=Kim \| first2=Panki \| ~~last3~~=Chen \| ~~first3~~=Zhen-Qing \| title=Global heat kernel estimates for symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-2011-05408-5 \| doi=10.1090/S0002-9947-2011-05408-5 \| year=2011 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=363 \| issue=9 \| pages=5021–5055}} </ref>		{{Citation \| last3=Kumagai \| first3=Takashi \| last2=Kim \| first2=Panki \| last1=Chen \| first1=Zhen-Qing \| title=Global heat kernel estimates for symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-2011-05408-5 \| doi=10.1090/S0002-9947-2011-05408-5 \| year=2011 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=363 \| issue=9 \| pages=5021–5055}} </ref>
	<ref name="BBCK09">		<ref name="BBCK09">
	{{Citation \| last1=Barlow \| first1=Martin T. \| last2=Bass \| first2=Richard F. \| last3=Chen \| first3=Zhen-Qing \| last4=Kassmann \| first4=Moritz \| title=Non-local Dirichlet forms and symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 \| doi=10.1090/S0002-9947-08-04544-3 \| year=2009 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=361 \| issue=4 \| pages=1963–1999}}		{{Citation \| last1=Barlow \| first1=Martin T. \| last2=Bass \| first2=Richard F. \| last3=Chen \| first3=Zhen-Qing \| last4=Kassmann \| first4=Moritz \| title=Non-local Dirichlet forms and symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 \| doi=10.1090/S0002-9947-08-04544-3 \| year=2009 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=361 \| issue=4 \| pages=1963–1999}}
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	{{Citation \| last1=Bass \| first1=Richard F. \| last2=Levin \| first2=David A. \| title=Transition probabilities for symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-02-02998-7 \| doi=10.1090/S0002-9947-02-02998-7 \| year=2002 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=354 \| issue=7 \| pages=2933–2953}}		{{Citation \| last1=Bass \| first1=Richard F. \| last2=Levin \| first2=David A. \| title=Transition probabilities for symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-02-02998-7 \| doi=10.1090/S0002-9947-02-02998-7 \| year=2002 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=354 \| issue=7 \| pages=2933–2953}}
	</ref>		</ref>
			<ref name="ChKu03">
			{{Citation \| last1=Chen \| first1=Zhen-Qing \| last2=Kumagai \| first2=Takashi \| title=Heat kernel estimates for stable-like processes on d-sets \| url=http://dx.doi.org/10.1016/S0304-4149(03)00105-4 \| doi=10.1016/S0304-4149(03)00105-4 \| year=2003 \| journal=Stochastic Processes and their Applications \| issn=0304-4149 \| volume=108 \| issue=1 \| pages=27–62}}</ref>
	}}		}}

imported>Moritz at 17:03, 5 July 2012

2012-07-05T17:03:22Z

@@ Line 23: / Line 23: @@
 More generally, the heat kernel can be shown to exists for nonlocal regular Dirichlet forms $(\mathcal{E}, D(\mathcal{E}))$. Assume
 \[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \]
-and $D(\mathcal{E}$  is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \|u\|_{L^2}^2$.
+and $D(\mathcal{E})$  is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \|u\|^2_{L^2}$.

imported>Moritz at 16:26, 5 July 2012

2012-07-05T16:26:43Z

← Older revision		Revision as of 11:26, 5 July 2012
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	For the special case $s=1/2$, the heat kernel coincides with the Cauchy kernel for the Laplace equation in the upper half space		For the special case $s=1/2$, the heat kernel coincides with the Cauchy kernel for the Laplace equation in the upper half space
	\[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]		\[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]

			More generally, the heat kernel can be shown to exists for nonlocal regular Dirichlet forms $(\mathcal{E}, D(\mathcal{E}))$. Assume
			\[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \]
			and $D(\mathcal{E}$ is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \\|u\\|_{L^2}^2$.

imported>Luis at 18:21, 28 April 2012

2012-04-28T18:21:19Z

← Older revision		Revision as of 13:21, 28 April 2012
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	In principle one could study the equation for any value of $s$. The values in the range $s \in (0,1]$ are particularly interesting because in that range the equation has a maximum principle.		In principle one could study the equation for any value of $s$. The values in the range $s \in (0,1]$ are particularly interesting because in that range the equation has a maximum principle.

			== Heat kernel ==
			The fractional heat kernel $p(t,x)$ is the fundamental solution to the fractional heat equation. It is the function which solves the equation
			\begin{align*}
			p(0,x) &= \delta_0 \\
			p_t(t,x) + (-\Delta)^s p &= 0
			\end{align*}

			The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t\|\xi\|^{2s}}$. There is no explicit formula in physical variables for general values of $s$, but the following inequalities are known to hold for some constant $C$
			\[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{\|x\|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{\|x\|^{n+2s}} \right). \]

			Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
			\[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]

			For the special case $s=1/2$, the heat kernel coincides with the Cauchy kernel for the Laplace equation in the upper half space
			\[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]

imported>Nestor at 21:38, 8 June 2011

2011-06-08T21:38:49Z

← Older revision		Revision as of 16:38, 8 June 2011
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	where $(-\Delta)^s$ stands for the [[fractional Laplacian]].		where $(-\Delta)^s$ stands for the [[fractional Laplacian]].

	In principle one could study the equation for any value of $s$. The values in the range $s \in (0,2]$ are particularly interesting because in that range the equation has a maximum principle.		In principle one could study the equation for any value of $s$. The values in the range $s \in (0,1]$ are particularly interesting because in that range the equation has a maximum principle.

imported>Luis: Created page with "The fractional heat equation refers to the parabolic equation \[ u_t + (-\Delta)^s u = 0,\] where $(-\Delta)^s$ stands for the fractional Laplacian. In principle one could s..."

2011-06-08T06:28:50Z

Created page with "The fractional heat equation refers to the parabolic equation \[ u_t + (-\Delta)^s u = 0,\] where $(-\Delta)^s$ stands for the fractional Laplacian. In principle one could s..."

New page

The fractional heat equation refers to the parabolic equation
\[ u_t + (-\Delta)^s u = 0,\]
where $(-\Delta)^s$ stands for the [[fractional Laplacian]].

In principle one could study the equation for any value of $s$. The values in the range $s \in (0,2]$ are particularly interesting because in that range the equation has a maximum principle.