I'm trying to find conditions on the symbol $q(x,\xi)$ of a Lévy-type process $X$ (d-dimensional) such that the following stopping time $\tau_u^{\alpha}$ is a.s. finite; $$ \tau^\alpha _u=\inf\left\{t\geq 0:\alpha^TX\geq u\right\},$$ where $\alpha$ is some element in $\mathbb{R}^d_{\geq 0}$ (better would be $\alpha \in\mathbb{R}^d$) and $u\in\mathbb{R}$.
With the long time behaviour result in [Bötcher, Schilling, Wang (2010) Lévy matters III] Theorem 5. 18, I have the answer for processes on a positive domain and for symmetric processes. But how do I approach this Problem in general?
Any hints to existing literature etc. are welcome.