Stories not complete at the end of the sprint

As per the scrum guide: 

The Definition of Done creates transparency by providing everyone a shared understanding of what work was completed as part of the Increment. If a Product Backlog item does not meet the Definition of Done, it cannot be released or even presented at the Sprint Review. Instead, it returns to the Product Backlog for future consideration.

I'm sure that somewhere in all of the Agile processes there is something.  But as far as Scrum goes, no there is not. There are a lot of other threads in this forum where this question has been asked so you might want to look at them to get some insights.

The usual common message is to that the Scrum framework has no process prescribed. The Scrum Team should discuss every instance when this happens to identify why it happened, what to do about the specific occurrence, and learn so that they may be able to prevent it from happening again. 

Is there an official Agile process for handing stories not complete at the end of a sprint?

I know some governments have tried, but I don't believe there is an "official Agile process" at all. I hope I'm right, because I'm not yet ready to trust officialdom with one.

In Scrum though, it's important for a team to ensure that the Product Backlog is kept up to date and re-estimated. It should reflect the total amount of work that is currently believed to remain. This includes any work that is left unfinished, i.e. which is not yet Done for any reason.

It's also important for a team to inspect and adapt their way of working, so they can make better forecasts about how much work they can complete within a time-box.

The Scrum guide mentions

> If a Product Backlog item does not meet the Definition of Done, it cannot be released or even presented at the Sprint Review. Instead, it returns to the Product Backlog for future consideration.

To me that implies that any incomplete PB item of a sprint needs to be carried over to the next sprint. In order for the team to bring the right amount of work into the sprint, the remaining work of the incomplete PB item needs to be re-estimated before it's brought into the sprint. Now some teams split the remaining work into a new story and close the old story, a practice that works well with keeping insights clean in story tracking platforms but adds the overhead of having to split up incomplete PB items, and IMO does not reflect fully the fact that the state of Done was not reached for that particular item.