Stochastic quintessence models: jerk and fine-structure variability
  constraints
  We report on constraints to the cosmological jerk parameter ($j$) and to
possible variability of the fine-structure constant ($\Delta \alpha/\alpha$)
based on stochastic quintessence models of dark energy, discussed by
Chongchitnan and Efstathiou (2007). We confirm the results by these authors in
the sense that many viable solutions can be obtained, obeying current
observational constraints in low redshifts. We add the observables $j$ and
$\Delta \alpha/\alpha$ to this conclusion. However, we find peculiarities that
may produce, in the nearby Universe, potential observational imprints in future
cosmological data. We conclude, for redshifts $z \lesssim 3$, that: {\it (i) }
$j(z)$ fluctuates due to the stochasticity of the models, reaching an amplitude
of up to $5\%$ relatively to the $\Lambda$CDM model value ($j_{\Lambda {\rm
CDM}}=1$); and {\it (ii)} by contrasting two distinct ("extreme") types of
solutions, variabilities in $\alpha(z)$, linked to a linear coupling ($\zeta$)
between the dark energy and electromagnetic sectors, are weakly dependent on
redshift, for couplings of the order $|\zeta| \sim 10 ^{-4}$, even for large
variations in the equation of state parameter at relatively low redshifts.
Nonlinear couplings produce an earlier and steeper onset of the evolution in
$\Delta \alpha/\alpha (z)$, but can still accommodate the data for weak enough
couplings.
