Title: The sum of log-normal variates in geometric Brownian motion,
Abstract: Geometric Brownian motion (GBM) is a key model for representing
self-reproducing entities. Self-reproduction may be considered the definition
of life [5], and the dynamics it induces are of interest to those concerned
with living systems from biology to economics. Trajectories of GBM are
distributed according to the well-known log-normal density, broadening with
time. However, in many applications, what's of interest is not a single
trajectory but the sum, or average, of several trajectories. The distribution
of these objects is more complicated. Here we show two different ways of
finding their typical trajectories. We make use of an intriguing connection to
spin glasses: the expected free energy of the random energy model is an average
of log-normal variates. We make the mapping to GBM explicit and find that the
free energy result gives qualitatively correct behavior for GBM trajectories.
We then also compute the typical sum of lognormal variates using Ito calculus.
This alternative route is in close quantitative agreement with numerical work.