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. In most cases we are interested in some subclass of linear operators. The simplest of all is the fractional Laplacian. We list several common simplifications below.
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 \[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
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.