  A multiplicative cascade can be thought of as a randomization of a measure on
the boundary of a tree, constructed from an iid collection of random variables
attached to the tree vertices. Given an initial measure with certain regularity
properties, we construct a continuous time, measure-valued process whose value
at each time is a cascade of the initial one. We do this by replacing the
random variables on the vertices with independent increment processes
satisfying certain moment assumptions. Our process has a Markov property: at
any given time it is a cascade of the process at any earlier time by random
variables that are independent of the past. It has the further advantage of
being a martingale and, under certain extra conditions, it is also continuous.
We discuss applications of this process to models of tree polymers and
one-dimensional random geometry.
