Do they mean the same?
Thank in advance,
go/run/flash etc through somebody's mind
if something goes through your mind, you have a thought, especially for a short time:
She knew what was going through his mind.
All kinds of questions ran through my mind.
After the accident, one of the things that went through my mind was whether I would be able to drive again.
on your/somebody's mind
a) if something is on your mind, you keep thinking or worrying about it:
He looked as though he had something on his mind.
Sorry I forgot. I've got a lot on my mind (=a lot of problems to worry about) at the moment.
b) if something is on your mind, that is what you are thinking about:
She's the type of person who just says what's on her mind.