I have the same question and have posted about this in https://forums.developer.apple.com/forums/thread/741939?page=1. Apple's documentation is a bit unclear about how the app is notified that a live activity is being created from push notification. Any further explanation will be very helpful.

Could anybody provide insight on how exactly the Live Activity is created via the push notification. Apple's docs state "When the system receives the ActivityKit push notification on a device, it starts a new Live Activity, wakes up your app, and grants it background run time to allow you to download assets that the Live Activity needs." Can anybody elaborate on how/where exactly this happens? I'm able to successfully obtain the pushToStartToken and can successfully deliver APNS payload via CloudKit - Push Notifications tool. I see in the delivery logs that my push notification was both received by the APNS server and successfully delivered to target device. Once delivered, I see my pushToStartTokenUpdates trigger once more and can get an updated token but, I don't fully understand how/where the live activity is actually created. Any information would be much appreciated