  The Brownian force model (BFM) is a mean-field model for the local velocities
during avalanches in elastic interfaces of internal space dimension $d$, driven
in a random medium. It is exactly solvable via a non-linear differential
equation. We study avalanches following a kick, i.e. a step in the driving
force. We first recall the calculation of the distributions of the global size
(total swept area) and of the local jump size for an arbitrary kick amplitude.
We extend this calculation to the joint density of local and global sizes
within a single avalanche, in the limit of an infinitesimal kick. When the
interface is driven by a single point we find new exponents $\tau_0=5/3$ and
$\tau=7/4$, depending on whether the force or the displacement is imposed. We
show that the extension of a single avalanche along one internal direction
(i.e. the total length in $d=1$) is finite and we calculate its distribution,
following either a local or a global kick. In all cases it exhibits a
divergence $P(\ell) \sim \ell^{-3}$ at small $\ell$. Most of our results are
tested in a numerical simulation in dimension $d=1$.
