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The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form

\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]

The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made.

== Absolutely continuous measure ==

In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.

We keep this assumption in all the examples below.

== Purely integro-differential operator ==

In this case we neglect the local part of the operator
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]

== Symmetric kernels ==
If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.

In the purely integro-differentiable case, it reads as
\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y. \]

The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x-y)-2u(x))$.

== Translation invariant operators ==
In this case, all coefficients are independent of $x$.
\[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]

== The fractional Laplacian ==

The [[fractional Laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.

\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]

== Stable operators ==

These are the operators whose kernel is homogeneous in $y$
\[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\]
They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.

== Uniformly elliptic of order $s$ ==

This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]

The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.

An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.

== Smoothness class $k$ of order $s$ ==
This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded
\[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]

== Order strictly below one ==

If a non symmetric kernel $K$ satisfies the extra local integrability assumption
\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
then the extra gradient term is not necessary in order to define the operator.

\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]

The modification in the integro-differential part of the operator becomes an extra drift term.

A uniformly elliptic operator of order $s<1$ satisfies this condition.

== Order strictly above one ==

If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
then the gradient term in the integral can be taken global instead of being cut off in the unit ball.

\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]

The modification in the integro-differential part of the operator becomes an extra drift term.

A uniformly elliptic operator of order $s>1$ satisfies this condition.

== More singular/irregular kernels ==

The concept of uniform ellipticity can be relaxed in various ways. The following family of operators was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />. The integro-differential operators have the form
\begin{align*}
\int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha > 1.
\end{align*}

For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \Omega \times \R^d \to \R$ is assumed to satisfy the following assumptions for all $x \in \Omega$,
* $K(x,h) \geq 0$ for all $h\in\R^d$.
This is a basic assumption for the integral operator to be a legit diffusion operator.

* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
This assumption is more general than $K(x,h) \leq (2-\alpha) \Lambda |h|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.

* For every $r>0$, there exists a set $A_r$ such that
** $A_r \subset B_{2r} \setminus B_r$.
** $A_r$ is symmetric in the sense that $A_r = -A_r$.
** $|A_r| \geq \mu |B_{2r} \setminus B_r|$.
** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|.
\]

This assumption says that the lower bound $K(x,h) \geq (2-\alpha) \lambda |h|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.

* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \]

This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.

== Indexed by a matrix ==
In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
\[ K_A(y) =  \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$. This property has been exploited in order to obtain an ABP estimate in integral form. <ref name="GS"/>

== Second order elliptic operators as limits of purely integro-differential ones ==

Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators
\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]
define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. We get
\[ \lim_{\sigma \to 2^-} L_\sigma u(x) = a_{ij} \partial_{ij} u(x),\]
where $a_{ij}$ is determined by the quadratic form
\[ a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.\]

A class of kernels that is big enough to recover all translation invariant elliptic operators of the form  $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels
\[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\]
where $A$ is an invertible symmetric matrix.

Conversely, the condition
\[ \lambda |e|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda |e|^2, \qquad \text{for all vectors } e \in \R^d,\]
defines the largest class of stable operators that may be considered uniformly elliptic. Indeed, this is the condition that ensures regularity of solutions to translation invariant integro-differential equations.<ref name="ros2014regularity" /> If we let the operator depend on $x$, it is an outstanding [[Open problems#Hölder estimates for singular integro-differential equations|open problem]] whether this condition alone (for every value of $x$) ensures [[Holder estimates]] for the integro-differential equation.

== Characterization via global maximum principle ==

A bounded linear operator 

\[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \] 

is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have

\[ (Lu)(x_0) \leq 0 \]

It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège <ref name="C65"/> <ref name="C64"/>: if $L$ satisfies the global maximum principle then it has the form

\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]

where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying

\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]

and $A(x),c(x)$ and $b(x)$ are bounded.


== See also ==

* [[Fractional Laplacian]]
* [[Levy processes]]
* [[Dirichlet form]]


== References ==
{{reflist|refs=
<ref name="C64">{{Citation | last1=Courrège | first1=Philippe | title=Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine | year=1964 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=88 | pages=3–30}}</ref>
<ref name="C65">{{Citation | last1=Courrège | first1=P. | title=Sur la forme intégro-différentielle des opéateurs de   $C_k^\infty(\mathbb{R}^n)$  dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum | journal=Sém. Théorie du potentiel (1965/66) Exposé | volume=2}}</ref>
<ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
<ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref>
<ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref>
}}