Cosmic ray propagation in the Universe in presence of a random magnetic field

The origin of the ultrahigh energy cosmic ray remains being a mystery. However, a considerable progress has been made in the past few years due to the good quality data recorded by current cosmic ray observatories. One of the recent achievements is obtaining firm observational evidence about the extragalactic origin of the most energetic cosmic rays by the Pierre Auger observatory. On the other hand, it is believed that there is a non-null turbulent magnetic field that fills the intergalactic medium. Therefore, the presence of the intergalactic magnetic field can play an important role on the propagation of the ultrahigh energy cosmic rays through the Universe, which in principle can be relevant to interpret the experimental data. In this work we present a system of partial differential equations that describes the propagation of the ultrahigh energy cosmic rays through the Universe, in the presence of a turbulent intergalactic magnetic field, that includes the diffusive and the ballistic regime of propagation and also the transition between them. Also, as an example of application, the system of equations is solved numerically in a simplified physical situation.