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Aims. Neutron stars (NSs) produced in the Milky Way are supposedly ten to theeighth - ten to the ninth, of which only $\sim 2 \times 10^{3}$ are observed.Constraining the phase space distribution of NSs may help to characterize theyet undetected population of stellar remnants. Methods. We perform Monte Carlosimulations of NS orbits, under different assumptions concerning the Galacticpotential and the distribution of progenitors and birth velocities. We studythe resulting phase space distributions, focusing on the statistical propertiesof the NS populations in the disk and in the solar neighbourhood. Results. Itis shown that $\sim 80$ percent of NSs are in bound orbits. The fraction of NSslocated in a disk of radius 20 kpc and width 0.4 kpc is $\lesssim 20$ percent.Therefore the majority of NSs populate the halo. Fits for the surface densityof the disk, the distribution of heights on the Galactic plane and the velocitydistribution of the disk, are given. We also provide sky maps of the projectednumber density in heliocentric Galactic coordinates (l, b). Our results arecompared with previous ones reported in the literature. Conclusions. Obviousapplications of our modelling are in the revisiting of accretion luminositiesof old isolated NSs, the issue of the observability of the nearest NS and theNS optical depth for microlensing events. These will be the scope of furtherstudies.

Galactic neutron stars I. Space and velocity distributions in the disk and in the halo