  Exchange-driven growth is a process in which pairs of clusters interact and
exchange a single unit of mass. The rate of exchange is given by an interaction
kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In
this paper we establish the fundamental mathematical properties of the mean
field kinetic equations of this process for the first time. We find two
different classes of behaviour depending on whether $K(j,k)$ is symmetric or
not. For the non-symmetric case, we prove global existence and uniqueness of
solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in
the sense that we show for a large class of initial conditions with kernels
satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On
the other hand, for symmetric kernels, we prove global existence of solutions
for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$
$\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$
($\beta>2).$ In the intermediate regime $3<\mu+\nu\leq4,$ we can only show
local existence. We conjecture that the intermediate regime exhibits
finite-time gelation in accordance with the heuristic results obtained for
particular kernels.
