Capsicum species, members of the Nightshade family and used throughout the world these days, are native to the Western hemisphere, and most probably originated from a common Mesoamerican ancestor.
An interesting bit of news: it seems that, over 6000 years ago, peoples living in what is now Ecuador made extensive use of these peppers in their cooking, mixing them with the staple maize. This 'paleobotanical' find was made by microscopically analyzing starch grains found at these ancient sites and discovering unique shapes characteristic to the Capsicum genus mixed in with the corn residues.
Given hot peppers' myriad uses for adjusting digestive function, reducing inflammation, impacting neurological symptoms, and more, it is not surprising that this medicine was domesticated so early. It has since spread across the globe. One of my favorite uses: externally, in a cream or salve, for reducing the pain and swelling associated with joint inflammation. This can be most easily made by steeping dried, chopped hot peppers in olive oil over a double boiler, for at least half an hour, straining, returning the infused oil to the double boiler and then thickening by adding beeswax (1oz beeswax by weight to 10oz of oil by volume). Wait until the beeswax melts completely, then pour the salve off into small jars for use. Be careful! Wash your hands with plenty of warm, soapy water after working with or using this salve, and don't forget to clean under your fingernails (alas, much personal experience has taught me this).