  Drift analysis aims at translating the expected progress of an evolutionary
algorithm (or more generally, a random process) into a probabilistic guarantee
on its run time (hitting time). So far, drift arguments have been successfully
employed in the rigorous analysis of evolutionary algorithms, however, only for
the situation that the progress is constant or becomes weaker when approaching
the target.
  Motivated by questions like how fast fit individuals take over a population,
we analyze random processes exhibiting a $(1+\delta)$-multiplicative growth in
expectation. We prove a drift theorem translating this expected progress into a
hitting time. This drift theorem gives a simple and insightful proof of the
level-based theorem first proposed by Lehre (2011). Our version of this theorem
has, for the first time, the best-possible near-linear dependence on $1/\delta$
(the previous results had an at least near-quadratic dependence), and it only
requires a population size near-linear in $\delta$ (this was super-quadratic in
previous results). These improvements immediately lead to stronger run time
guarantees for a number of applications.
  We also discuss the case of large $\delta$ and show stronger results for this
setting.
