There are a few equivalent definitions for the fractional Laplacian operator for a constant . One is given by
and the other one is given by
where and are normalizing constants and is a symmetric Levy measure on . In this below, we will show they are actually given equivalent with .
For simplicity, we first use the following notions. We denote
Let’s try to show the fact in a little bit more general case for .
If is an even function, then and for any function to make and well defined.
Proof: Indeed, we have the symmetry of , i.e. . Therefore, we obtain
The last equality in the above holds by adding two representations of given above. Similarly, we write by
and adding up two representations of to get .
Let and be two constants. If is a -stable process with , then is . .
Proof: We denote and and try to show . For any , we have, with
The proof is accomplished by noting that
In the above, is the dimension of , and we used change of variable by with the facts and .