  We introduce a description of the collective transverse dynamics of charged
(proton) beams in the stability regime by suitable classical stochastic
fluctuations. In this scheme, the collective beam dynamics is described by
time--reversal invariant diffusion processes deduced by stochastic variational
principles (Nelson processes). By general arguments, we show that the diffusion
coefficient, expressed in units of length, is given by $\lambda_c\sqrt{N}$,
where $N$ is the number of particles in the beam and $\lambda_c$ the Compton
wavelength of a single constituent. This diffusion coefficient represents an
effective unit of beam emittance. The hydrodynamic equations of the stochastic
dynamics can be easily recast in the form of a Schr\"odinger equation, with the
unit of emittance replacing the Planck action constant. This fact provides a
natural connection to the so--called ``quantum--like approaches'' to beam
dynamics. The transition probabilities associated to Nelson processes can be
exploited to model evolutions suitable to control the transverse beam dynamics.
In particular we show how to control, in the quadrupole approximation to the
beam--field interaction, both the focusing and the transverse oscillations of
the beam, either together or independently.
