Lagrangian forward and backward models are introduced into a coarse-grid ocean global circulation model to trace the ventilation routes of the deep North Pacific Ocean. The random walk aspect in the Lagrangian model is dictated by a rotated isopycnal diffusivity tensor in the circulation model, and the effect of diffusion is explicitly resolved by means of stochastic terms in the Lagrangian model. The analogy between the probability distribution of a Lagrangian model with Green's function of an Eulerian tracer transport equation is established. The estimated first- and last-passage time density of the deep North Pacific using both the Eulerian and the Lagrangian models ensured that the Lagrangian pathways and their ensemble statistics are consistent with the Eulerian tracer transport and its adjoint model. Moreover, the sample pathways of the ventilated mass fractions of the deep North Pacific particles to and from the ocean surface are studied.@en