  We investigate statistical properties of trails formed by a random process
incorporating aggregation, fragmentation, and diffusion. In this stochastic
process, which takes place in one spatial dimension, two neighboring trails may
combine to form a larger one and also, one trail may split into two. In
addition, trails move diffusively. The model is defined by two parameters which
quantify the fragmentation rate and the fragment size. In the long-time limit,
the system reaches a steady state, and our focus is the limiting distribution
of trail weights. We find that the density of trail weight has power-law tail
$P(w) \sim w^{-\gamma}$ for small weight $w$. We obtain the exponent $\gamma$
analytically, and find that it varies continuously with the two model
parameters. The exponent $\gamma$ can be positive or negative, so that in one
range of parameters small-weight tails are abundant, and in the complementary
range, they are rare.
