Is there any way to start a process on a linux computer if I have ftp connection or network connection?
Not with FTP, no. FTP is for transferring files. If you could launch remote processes it wouldn't be called File Transfer Protocol.
With a network connection you can do anything. That's what SSH is for (it launches a remote shell where you can run commands). You could also program any kind of specific listener you want using your own ports (warning, security woes ahead!).
You will have to be more specific with what you are trying to accomplish if you want a more specific answer.