Fallow and Axis or Chital deer are originally Asian as Christer suggests.

Fallow deer originated in Mesopotamia and Persia. There is a Persian sub-species today in very limited numbers. How they spread from Mesopotamia is anyone's guess but the Roman's during the Roman Empire era spread them to many new locations including Britian where red deer were native. Probably also to France, Spain, Italy and North Africa.

Chital or Axis deer originate from the Indian sub-continent and were spread by acclimatisation societies during the British Empire days to a number of locations too such as Australia. Fallow deer too were spread during this time successfully to many places.

Since then of course deer farmers have also spread the species and also safari operators.

I will see what Whitehead's "Deer Species of the World" has to say on the topic and correct or add to anything above.

Both species, the menil coloured fallow and chital are in my opinion the loveliest deer species due to their white spotted russett coloured coats.