> Is there historical evidence kept by the Persian Empire and the Magi concerning the birth of Messiah
If it ever existed, it doesn't now. At least it's never been found.
The only such records that survived are: It is recorded in antiquity that magians arrived at other notable historical moments. Seneca (
Epistle 58.31 ) tells of magians who offered sacrifices in Athens upon knowledge of the death of Plato. Cicero (
De Divin. i.47 ) refers to the constellation from which, on the birth night of Alexander the Great, magians foretold that the curse of Asia was born.
But Seneca wrote his account 400 years after Plato was alive; we don't have the Persian account. Cicero was 200 years after Alexander, so the same thing—we have none of the original Persian records. Maybe Seneca and Cicero had access to original records; we do not.
Seneca:
"You know, I am sure, that Plato had the good fortune, thanks to his careful living, to die on his birthday, after exactly completing his eighty-first year. For this reason, wise men of the East, who happened to be in Athens at that time, sacrificed to him after his death, believing that his length of days was too full for a mortal man, since he had rounded out the perfect number of nine times nine. I do not doubt that he would have been quite willing to forgo a few days from this total, as well as the sacrifice."
Cicero:
"It certainly must be true that even barbarians have some power of foreknowledge and of prophecy, if the following story of Callanus of India be true: As he was about to die and was ascending his funeral pyre, he said: ‘What a glorious death! The fate of Hercules is mine. For when this mortal frame is burned the soul will find the light.' When Alexander directed him to speak if he wished to say anything to him, he answered: 'Thank you, nothing, except that I shall see you very soon. So it turned out, for Alexander died in Babylon a few days later. I am getting slightly away from dreams, but I shall return to them in a moment. Everybody knows that on the same night in which Olympias was delivered of Alexander the temple of Diana at Ephesus was burned, and that the magi began to cry out as the day was breaking: 'Asia's deadly curse was born last night.' But enough of Indians and magi."