Alot of questions

I was recently diagnosed with HSV 2 and I have alot of questions.My girlfriend was diagnosed first and I decided to go get tested.I showed no symptoms at all and the doctor told me that maybe I was asymptomatic and that I might not ever show signs.Well to make this story short I requested Valtrex even though I showed no outward physical signs.She didnt quite agree with treating something she said didnt exist but I told her that my biggest issue was suppression and lowering the chances of transmission.My first question is how likely am I to transmit to future sexual partners if I am asymptomatic amd taking suppression medication?