The idea of recasting well-posedness problems for non-linear
parabolic-type PDE in terms of averages of associated multiplicative
cascades goes back to Le Jan and Sznitman's 1997 paper on Navier-Stokes
equation (NSE). More recently, in collaboration with N. Michalowski, E.
Waymire, and E. Thomann, we looked into the connection between uniqueness of
self-symilar (symmetry-preserving) solutions of 3D NSE and uniqueness for
general solutions. In this work we explore these ideas on a much simpler
case of complex Burgers equations. On one hand the probabilistic description
is more explicit allowing us to illustrate the use of such techniques to
study both global existence and uniqueness in a wide scaling-invariant space
as well as to study limit behavior of the solutions. On the other hand, the
multiplicative process itself is non-trivial and further attempts to
simplify it leads to a family of "delayed Yule" processes that may
be of independent interest.