  Consider n non-intersecting particles on the real line (Dyson Brownian
motions), all starting from the origin at time=0, and forced to return to x=0
at time=1. For large n, the average mean density of particles has its support,
for each 0<t<1, within the interior of an ellipse. The Airy process is defined
as the motion of these non-intersecting Brownian motions for large n, but
viewed from an arbitrary point on the ellipse with an appropriate space-time
rescaling. Assume now a finite number r of these particles are forced to a
different target point. Does it affect the Brownian fluctuations along the
ellipse for large n? In this paper, we show that no new process appears as long
as one considers points on the ellipse, for which the t-coordinate is smaller
than the t-coordinate of the point of tangency of the tangent to the curve
passing through the target point. At this point of tangency the fluctuations
obey a new statistics: the Airy process with r outliers (in short: {\bf r-Airy
process}). The log of the transition probability of this new process is given
by the Fredholm determinant of a new kernel (extending the Airy kernel) and it
satisfies a non-linear PDE in x and the time.
