We study the stellar and wind properties of a sample of Galactic O dwarfs to track the conditions under which weak winds (i.e. mass loss rates lower than 10^-8 M_&sun; yr^-1) appear. The sample is composed of low and high luminosity dwarfs including Vz stars and stars known to display qualitatively weak winds. Atmosphere models including non-LTE treatment, spherical expansion and line blanketing are computed with the code CMFGEN (Hillier & Miller 1998, ApJ, 496, 407). Both UV and Halpha lines are used to derive wind properties while optical H and He lines give the stellar parameters. We find that the stars of our sample are usually 1 to 4 Myr old. Mass loss rates of all stars are found to be lower than expected from the hydrodynamical predictions of Vink et al. (2001, A&A, 369, 574). For stars with log {L}/{L_&sun;} ⪆ 5.2, the reduction is by less than a factor 5 and is mainly due to the inclusion of clumping in the models. For stars with log {L}/{L_&sun;} ⪅ 5.2 the reduction can be as high as a factor 100. The inclusion of X-ray emission (possibly due to magnetic mechanisms) in models with low density is crucial to derive accurate mass loss rates from UV lines, while it is found to be unimportant for high density winds. The modified wind momentum luminosity relation shows a significant change of slope around this transition luminosity. Terminal velocities of low luminosity stars are also found to be low. Both mass loss rates and terminal velocities of low L stars are consistent with a reduced line force parameter alpha. However, the physical reason for such a reduction is still not clear although the finding of weak winds in Galactic stars excludes the role of a reduced metallicity. There may be a link between an early evolutionary state and a weak wind, but this has to be confirmed by further studies of Vz stars. X-rays, through the change in the ionisation structure they imply, may be at the origin of a reduction of the radiative acceleration, leading to lower mass loss rates. A better understanding of the origin of X-rays is of crucial importance for the study of the physics of weak winds.