In this paper, Hunt's hypothesis (H) and Getoor's conjecture for L\'{ e}vy processes are revisited. Let $X$ be a L\'{ e}vy process on $\mathbf{ R}^n$ with L\'{ e}vy-Khintchine exponent $(a,A,\mu)$. { First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $\mu({ \mathbf{ R}^n\backslash \sqrt{ A}\mathbf{ R}^n})<\infty$, we show that $X$ satisfies (H) if and only if the equation $$ \sqrt{ A}y=-a-\int_{ \{ x\in { \mathbf{ R}^n\backslash \sqrt{ A}\mathbf{ R}^n}:\,|x|<1\}}x\mu(dx),\ y\in \mathbf{ R}^n, $$ has at least one solution. Finally, we show that if $X$ is a subordinator and satisfies (H) then its drift coefficient must be 0.}
Categories: math.PR