The 'provider' is the driver required to allow SQL Server to talk to the remote server. These are normally installed pieces of software, so can't simply be copied between machines.
While not being overly familiar with PervasiveDB, I would assume that like most other manufacturers drivers, it will be included in their client release, which is available for download here - http://www.pervasivedb.com/psqlv11/pages/default.aspx
Once you've installed this, then you can build your linked server connection. Within SSMS you can right click on your existing Linked Server and make a create script which you can use on your SQL2012 instance. Though be warned that this won't copy all the security details (passwords in particular), so you may need to look this up in the documentation, or have a new user created in the target database.
Sometimes you can have issues with different versions of client software, differing client setup or not getting the permissions identical on the rebuild. You can avoid this in future (or make someone's life a lot easier in the future) by documenting everything as you set it up.