  We generalize a model of growth over a disordered environment, to a large
class of It\=o processes. In particular, we study how the microscopic
properties of the noise influence the macroscopic growth rate. The present
model can account for growth processes in large dimensions, and provides a bed
to understand better the trade-off between exploration and exploitation. An
additional mapping to the Schr\"ordinger equation readily provides a set of
disorders for which this model can be solved exactly. This mean-field approach
exhibits interesting features, such as a freezing transition and an optimal
point of growth, that can be studied in details, and gives yet another
explanation for the occurrence of the $\textit{Zipf law}$ in complex,
well-connected systems.
