Fractional heat equation

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where $(-\Delta)^s$ stands for the [[fractional Laplacian]].
where $(-\Delta)^s$ stands for the [[fractional Laplacian]].
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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.
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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.
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== Heat kernel ==
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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
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\begin{align*}
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p(0,x) &= \delta_{\{x\}} \\
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p_t(t,x) + (-\Delta)^s p &= 0
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\end{align*}
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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 $C$
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\[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
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Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
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\[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]
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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
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\[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]
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More generally, the heat kernel can be shown to exists for certain nonlocal regular Dirichlet forms $(\mathcal{E}, D(\mathcal{E}))$. Assume
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\[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \]
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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"/>.
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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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== References ==
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{{reflist|refs=
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<ref name="CKK11">
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{{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>
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<ref name="BBCK09">
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{{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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</ref>
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<ref name="BL02">
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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}}
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</ref>
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<ref name="ChKu03">
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{{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>
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}}

Latest revision as of 21:55, 8 July 2012

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 $s \in (0,1]$ 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_{\{x\}} \\ 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 \[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]

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

Then the corresponding transition semigroup has a heat kernel $p(t,x,y)$ under quite general assumptions on $J(x,y)$[1].

If $J(x,y)$ is comparable to $|x-y|^{-d-\alpha}$, $p(t,x,y)$ satisfies a bound like above [2][3]. One can relax the assumptions significantly and still prove sharp bounds for small time as well as for large time [4].


References

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