Where is the lightning history persisted?
We like to say that Bitcoin is decentralized, and it is, but all the data can be viewed as having a centralized "source of truth": the blockchain.  All transactions that have ever occurred are a part of this source of truth.  The management and upkeep of this source of truth, however, is decentralized.  Nobody is in charge of it, yet everybody agrees on a single set of data.

The lightning network, however, is even more decentralized.  It doesn't just manage and update itself without central authority, but there is no agreed upon "truth" of the network.  The activity that occurs between two participants in a channel is only the business of the participants, and nobody else ever needs to see it.  Therefore it is not shared, and unless the participants want to, it is not stored.  Once a channel closes, the participants have completely settled all transactions between each other with a single transaction to the main Bitcoin blockchain.  They may very well persist the activity data for their own purposes, but it is in no way required for them to continue to participate.