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
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
Proof: This is the consequence of Taylor expansion.
Exercise 1 Find .
Now we can rewrite , where
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, .
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.