It's off chain. Lightning transactions go solely over the lightning channels. On chain fees are only incurred, and paid, for on chain transactions, which, with regard to Lightning, only occur when money enters the Lightning network, or leaves it. As long as it's lightning to lightning, there are no fees paid to miners. There may be smaller fees paid to the lightning network, but in my experience these are usually either zero or a couple of sats at this time. 

 yeah, that's why I asked because then who is securing the network? If the miners aren't being rewarded for the difficulty, on -chain then how does it actually stay secure and online in the long term