  Noisy dynamical models are employed to describe a wide range of phenomena.
Since exact modeling of these phenomena requires access to their microscopic
dynamics, whose time scales are typically much shorter than the observable time
scales, there is often need to resort to effective mathematical models such as
stochastic differential equations (SDEs). In particular, here we consider
effective SDEs describing the behavior of systems in the limits when natural
time scales became very small. In the presence of multiplicative noise (i.e.,
noise whose intensity depends upon the system's state), an additional drift
term, called noise-induced drift, appears. The nature of this noise-induced
drift has been recently the subject of a growing number of theoretical and
experimental studies. Here, we provide an extensive review of the state of the
art in this field. After an introduction, we discuss a minimal model of how
multiplicative noise affects the evolution of a system. Next, we consider
several case studies with a focus on recent experiments: Brownian motion of a
microscopic particle in thermal equilibrium with a heat bath in the presence of
a diffusion gradient, and the limiting behavior of a system driven by a colored
noise modulated by a multiplicative feedback. This allows us to present the
experimental results, as well as mathematical methods and numerical techniques
that can be employed to study a wide range of systems. At the end we give an
application-oriented overview of future projects involving noise-induced
drifts, including both theory and experiment.
