  We present a short review of possible applications of the Wheeler-De Witt
equation to cosmological models based on the low-energy string effective
action, and characterised by an initial regime of asymptotically flat, low
energy, weak coupling evolution. Considering in particular a class of
duality-related (but classically disconnected) background solutions, we shall
discuss the possibility of quantum transitions between the phases of pre-big
bang and post-big bang evolution. We will show that it is possible, in such a
context, to represent the birth of our Universe as a quantum process of
tunneling or "anti-tunneling" from an initial state asymptotically approaching
the string perturbative vacuum.
