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Let $Z_t$ be a Levy process with the generator $A$, and consider the following SDE: $$ dX_t=f(X_t)dt+dZ_t, $$ may I know can I represent the generator $L$ of $X$ in terms of $A$?

This gives if $f(X_t)=BX_t$, the generator of $X$ is given by $$ Lu=Au+Bx\cdot \nabla u. $$ Then can I conjecture that $$ Lu=Au+f(x)\cdot \nabla u, $$ for more general $f$? Could you provide some ideas about how to approach the problem or some references on this topic?

John
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    Yes, that's correct - at least for functions $u$ which are sufficiently regular. For instance if $u$ is twice differentiable, you can prove the assertion by applying Itô's formula. – saz Aug 23 '17 at 08:41
  • @saz Do you mean I can establish the desired result for Lipschitz $f$ by mimicking the process of identifying the generator of Ito-diffusions? – John Aug 23 '17 at 08:53
  • Yes, the main difference is that you need Itô's formula for jump processes (and not only for Itô diffusions). – saz Aug 23 '17 at 08:56
  • @saz Thank you. However, I was wondering where I consider $\frac{E^x[f(X_t)]-f(x)}{t}$ as $t\to 0$, do I need to consider the usual pointwise convergence or the uniform convergence? – John Aug 23 '17 at 11:48
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    You can either a) show uniform convergence or b) show pointwise convergence and that the limit function $Lu$ is continuous and vanishes at infinity. – saz Aug 23 '17 at 14:10

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