  We describe the transverse beam distribution in particle accelerators within
the controlled, stochastic dynamical scheme of the Stochastic Mechanics (SM)
which produces time reversal invariant diffusion processes. This leads to a
linearized theory summarized in a Shchr\"odinger--like (\Sl) equation. The
space charge effects have been introduced in a recent paper~\cite{prstab} by
coupling this \Sl equation with the Maxwell equations. We analyze the space
charge effects to understand how the dynamics produces the actual beam
distributions, and in particular we show how the stationary, self--consistent
solutions are related to the (external, and space--charge) potentials both when
we suppose that the external field is harmonic (\emph{constant focusing}), and
when we \emph{a priori} prescribe the shape of the stationary solution. We then
proceed to discuss a few new ideas~\cite{epac04} by introducing the generalized
Student distributions, namely non--Gaussian, L\'evy \emph{infinitely divisible}
(but not \emph{stable}) distributions. We will discuss this idea from two
different standpoints: (a) first by supposing that the stationary distribution
of our (Wiener powered) SM model is a Student distribution; (b) by supposing
that our model is based on a (non--Gaussian) L\'evy process whose increments
are Student distributed. We show that in the case (a) the longer tails of the
power decay of the Student laws, and in the case (b) the discontinuities of the
L\'evy--Student process can well account for the rare escape of particles from
the beam core, and hence for the formation of a halo in intense beams.

Accelerator Physics;Statistical Mechanics;Plasma Physics;Quantum Physics