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| The fractional heat equation refers to the parabolic equation | | The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great generality of assumptions. |
| \[ u_t + (-\Delta)^s u = 0,\]
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| 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 $s \in (0,1]$ are particularly interesting because in that range the equation has a maximum principle.
| | The comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the [[maximum principle]]. The uniqueness of the solution of the equation is an immediate consequence. |
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| == Heat kernel == | | == General statement == |
| 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 $C$ | | The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions. |
| \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
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| Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
| | Other boundary conditions require appropriate modifications. |
| \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]
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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
| | === Elliptic case === |
| \[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]
| | We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true. |
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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
| | Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that |
| \[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \]
| | $Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in an open domain $\Omega$, and |
| and $D(\mathcal{E})$ is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \|u\|^2_{L^2}$.
| | $u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well. |
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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"/>.
| | === Parabolic case === |
| | We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true. |
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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"/>.
| | Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that |
| | | $Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $(0,T] \times \Omega$, and |
| | | $u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well. |
| == References ==
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| {{reflist|refs= | |
| <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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| }}
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The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great generality of assumptions.
The comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. The uniqueness of the solution of the equation is an immediate consequence.
General statement
The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions.
Other boundary conditions require appropriate modifications.
Elliptic case
We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.
Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that
$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in an open domain $\Omega$, and
$u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.
Parabolic case
We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.
Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that
$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $(0,T] \times \Omega$, and
$u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.