I read Sato's section 21 in 'Lévy processes and infinitly divisible distributions (1999)' and I don't understand this one passage which is a consequence of Theorem 21.5.
It reads: 'A consequence of Theorem $21.5$ should be contemplated. A Lévy process on $\mathbb{R}$ generated by $(A,\nu, \gamma)$ with $A=0$, $\nu((-\infty,0))=0$, and $\int_{(0,1]}x\nu(dx)=\infty$ has positive jumps only, does not have a Brownian-like part, but it is fluctuating, not increasing, no matter how large $\gamma$ is. Moreover, it is not increasing in any time interval (Theorem 21.9(ii)). An explanation is that such a process can exist only with infinitely strong drift in the negative direction, which cancels the divergence of the sum of jumps. It causes a random continuous motion in the negative direction.'
Intuition of Lévy process with infinite negative drift.
Why does the process have to be fluctuating and what can we deduct from the variation. Why can't it be decreasing and what does the variation have to do with this property. I also don't understand how $\int_{(0,1]}x\nu(dx)=\infty$ specifaically has this effect (only that it entails that the Lévy process is not increasing). Could someone provide me with a thourough explanation and maybe even some illustrative graphs?