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.

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

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.

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 $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{s-1} u \right] (x)$ 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.