It is likely that this rash is simply irritation caused by friction from the bra and excessive sweating. The good news is that it is not spreading and it is not itchy or painful.
I would suggest, in addition to good skin hygiene obviously, to try applying a barrier skin protectant while running to the involved areas. You could try a thick ointment like hydrolated petrolatum or there are several specialty barrier products that you could purchase in a running store. You should apply these before running to minimize the irritation. Also, upon returning from your run, you should immediately shower to remove the irritating debris that accumulates on the skin from sweating. After showering, apply a skin moisturizer. If this does not clear things up, you may want to talk to your primary care doctor
. Some of these rashes do require a steroid cream in order to clear up completely, and they can help you make that decision.
One other possibility, which seems less likely, is that you are developing a contact allergy to something in the sports bra, such as the elastic. Contact allergies tend to be more itchy, but not always. If you think this is a possibility, switching to a different brand or type of bra might be helpful.