If the Persians conquered Greece how would the existence/expansion of the Roman Empire?

[Historyman 14]

The Roman Empire lasted for about 4 centuries (14 if counting Byzantium), what's not to say that Persia couldn't pull the same feat off.

Even before Alexander came around, Persia was already on a decline.

The Persian Empire was weakening for generations before Alexander show up, more and more they were dependent on mercenaries to fight their wars (mainly Greek mercenaries at that), their dynastic politics were absolutely toxic and the satraps were growing increasingly independent-minded, and there'd been a series of major revolts already.

I think we often place too much on Persia to get stuff done and everything work out for them. Adding Hellas would be just another drain and waste for them. (Like Al Gore 2000 winning solving all the world's woes, or Bobby Kennedy doing everything, but walking on water, or the Song pulling a industrial revolution.)

The Achaemenids may have another generation in them, at least, but you very likely see Egypt, Greece, and the Satrpas in the East revolt and gain their freedom while a new dynasty takes over. Losing Egypt and Greece may very well be for the better.

I think that Rome will fail to rise to become a challenger to Persia. There will be many dissidents fleeing from Greece to southern Italy and further aid from the Greeks could strengthen the influence of the Greeks in Italy leading to Rome being killed in its cradle. Maybe the Carthaginians would get aid from the Persians in battles against the Greeks in the western Mediterranean and thus when Rome comes knocking on their door, the Carthaginians are better equipped to defeat the Romans.

I don't see why Persia and Carthage would do that. They won't bug each other. Carthage would do its own thing.