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

and

Let’s try to show the fact in a little bit more general case for .

**Proposition 1** * *

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 .

**Example 1** * *

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 .

### Like this:

Like Loading...