Let $A$ be a pseudo-differential operator with symbol $a(x,\xi)=e^{-x}\psi(\xi)$ where $\psi$ is a continuous negative definite function. It is known that $A$ is the generator of a self-similar Markov process that has $e^x\,dx$ as an excessive measure. If we denote the corresponding Markov semi-group by $(P_t)_{t\ge 0}$, it has a natural extension on the weighted Hilbert space $L^2(\mathbb{R}, e^x\,dx)$. My question is the following:
Can we determine the domain of the generator of $(P_t)_{t\ge 0}$ in $L^2(\mathbb{R}, e^x\,dx)$? In other words, can we determine the $L^2(\mathbb{R}, e^x\,dx)$ extension of $A$?
An update: Also assume that $\psi$ is smooth. Can we find such a $\psi$ (which should also be negative definite) so the the operator $A:L^2(\mathbb{R}, e^x\,dx)\to L^2(\mathbb{R}, e^x\,dx)$ is bounded?