  An exact description of the statistical motion of active particles in three
dimension is presented in the framework of a generalized diffusion equation.
Such a generalization contemplates a non-local, in time and space, connecting
(memory) function. This couples the rate of change of the probability density
of finding the particle at position $\boldsymbol{x}$ at time $t$, with the
Laplacian of the probability density at all previous times and to all points in
space. Starting from the standard Fokker-Planck-like equation for the
probability density of finding an active particle at position $\boldsymbol{x}$
swimming along the direction $\hat{\boldsymbol{v}}$ at time $t$, we derive in
this paper, in an exact manner, the connecting function that allows a
description of active motion in terms of this generalized diffusion equation.
