Stochastic modeling of star-formation histories I: the scatter of the
  star-forming main sequence
  We present a framework for modelling the star-formation histories of galaxies
as a stochastic process. We define this stochastic process through a power
spectrum density with a functional form of a broken power-law. Star-formation
histories are correlated on short timescales, the strength of this correlation
described by a power-law slope, $\alpha$, and they decorrelate to resemble
white noise over a timescale that is proportional to the timescale of the break
in the power spectrum density, $\tau_{\rm break}$. We use this framework to
explore the properties of the stochastic process that, we assume, gives rise to
the log-normal scatter about the relationship between star-formation rate and
stellar mass, the so-called galaxy star-forming main sequence. Specifically, we
show how the measurements of the normalisation and width ($\sigma_{\rm MS}$) of
the main sequence, measured in several passbands that probe different
timescales, give a constraint on the parameters of the underlying power
spectrum density. We first derive these results analytically for a simplified
case where we model observations by averaging over the recent star-formation
history. We then run numerical simulations to find results for more realistic
observational cases. As a proof of concept, we use observational estimates of
the main sequence scatter at $z\sim0$ and $M_{\star}\approx10^{10}~M_{\odot}$
measured in H$\alpha$, UV+IR and the u-band, and show that combination of these
point to $\tau_{\rm break}=178^{+104}_{-66}$ Myr, when assuming $\alpha=2$.
This implies that star-formation histories of galaxies lose "memory" of their
previous activity on a timescale of $\sim200$ Myr, highlighting the importance
of baryonic effects that act over the dynamical timescales of galaxies.
