It duplicates the spell. Therefore, there is no cast involved, and, if i remember well, arcane blast stack is applied on the cast, not on the spell itself.
Think of the proc being similar to Windfury from a shaman. They don't have to do anything and it just happens. There's no bonus to the "attack" for the proc but in all other aspects its identical to it except its just 1 extra "attack" not 3. This includes any dots you have going and what is assumed any other procs you get (hurricane proc etc). Its simply 1 extra damage dealing of 1 of your attacks.