  We provide a new construction of the Brownian disks, which have been defined
by Bettinelli and Miermont as scaling limits of quadrangulations with a
boundary when the boundary size tends to infinity. Our method is very similar
to the construction of the Brownian map, but it makes use of the positive
excursion measure of the Brownian snake which has been introduced recently.
This excursion measure involves a random continuous tree whose vertices are
assigned nonnegative labels, which correspond to distances from the boundary in
our approach to the Brownian disk. We provide several applications of our
construction. In particular, we prove that the uniform measure on the boundary
can be obtained as the limit of the suitably normalized volume measure on a
small tubular neighborhood of the boundary. We also prove that connected
components of the complement of the Brownian net are Brownian disks, as it was
suggested in the recent work of Miller and Sheffield. Finally, we show that
connected components of the complement of balls centered at the distinguished
point of the Brownian map are independent Brownian disks, conditionally on
their volumes and perimeters.
