You may own whatever you want in your home country. The J-1 only restricts your ability to work here, not anywhere else in the world. You could manage your whole operations outside the U.S. from your computer in the U.S. and this would not violate your J.
But of course. One thing does not conflict with the other...
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
You can own a business in your home country. However, you cannot work for that business while you're inside the United States training on a J1 visa.
This is general information, not legal advice, and does not create an attorney client relationship.