  Quantum dynamical localization occurs when quantum interference stops the
diffusion of wave packets in momentum space. The expectation is that dynamical
localization will occur when the typical transport time of the momentum
diffusion is greater than the Heisenberg time. The transport time is typically
computed from the corresponding classical dynamics. In this paper, we present
an alternative approach based purely on the study of spectral fluctuations of
the quantum system. The information about the transport times is encoded in the
spectral form factor, which is the Fourier transform of the two-point spectral
autocorrelation function. We compute large samples of the energy spectra (of
the order of $10^6$ levels) and spectral form factors of 22 stadium billiards
with parameter values across the transition between the localized and extended
eigenstate regimes. The transport time is obtained from the point when the
spectral form factor transitions from the non-universal to the universal regime
predicted by random matrix theory. We study the dependence of the transport
time on the parameter value and show the level repulsion exponents, that are
known to be a good measure of dynamical localization, depend linearly on the
transport times obtained in this way.
