In this below, we will discuss some connections on the several representations on the generator of Levy process.

For simplicity, we only discuss one dimensional Levy measure. Throughout this discussion, we assume is a Levy measure on , i.e. a non-negative measure satisfying

Let’s consider an operator given by

The above (1) is often given as the generator of a purely non-gaussian Levy processes in the probability literature. Another operator used in [Caffarelli and Silvestre 09] is

The main goal of this discussion is to show that (1) and (2) are equivalent whenever is symmetric, but not in general.

**Proposition 1** * is a subset of the domain of , i.e. . *

*Proof:* This is the consequence of Taylor expansion.

**Exercise 1** * Find . *

Now we can rewrite , where

and

**Lemma 2** * for all . *

*Proof:* By Taylor expansion, one can write

where the function is a non-negative monotone function with . Furthermore, since is a CDF of a probability measure on , is right continuous. Therefore, .

**Proposition 3** * If the Levy measure is symmetric, i.e. for all Borel sets , then for all . *

*Proof:* It’s equivalent to show that . Note for any , we can write

The last term of the above equation is zero due to the symmetry of , and it yields

Finally, one can take the limit as and apply Lemma 2 to conclude the conclusion.

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