# Linear integro-differential operator

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 $y$ $K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.$ They are the generators of stable Lévy processes.

## 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 kernels was considered in the paper of Silvestre and Schwab.[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 $a_{ij}$.

## 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. 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.

## 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 [2] [3]: 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.

2. Courrège, P., "Sur la forme intégro-différentielle des opéateurs de $C_k^\infty(\mathbb{R}^n)$ dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum", Sém. Théorie du potentiel (1965/66) Exposé 2
3. Courrège, Philippe (1964), "Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine", Bulletin des Sciences Mathématiques. 2e Série 88: 3–30, ISSN 0007-4497