  We present an analysis of the long-term optical variability for $\sim50,000$
nearby (z<0.055) galaxies from the NASA-Sloan Atlas, $35,000$ of which are
low-mass ($M_{\ast}<10^{10}~M_{\odot}$). We use difference imaging of Palomar
Transient Factory (PTF) R-band observations to construct light curves with
typical baselines of several years. We then search for subtle variations in the
nuclear light output. We determine whether detected variability is AGN-like by
assessing the fit quality to a damped random walk model. We identify 424
variability-selected AGN, including 244 with stellar masses between $10^{7}$
and $10^{10}~M_{\odot}$. 75% of low-mass galaxies with AGN-like variability
have narrow emission lines dominated by star formation. After controlling for
nucleus magnitude, the fraction of variable AGN is constant down to
$M_{\ast}=10^{9}~M_{\odot}$, suggesting no drastic decline in the BH occupation
fraction down to this stellar mass regime. Combining our NASA-Sloan Atlas
sample with samples of nearby galaxies with broad H$\alpha$ emission, we find
no dependence of variability properties with black hole mass. However, we
caution that the variable AGN fraction is strongly dependent on baseline. For
baselines less than two years, the variable fraction for the full sample is
0.25%, compared to 1.0% for baselines longer than two years. Finally, comparing
Stripe 82 light curves (Baldassare et al. 2018) to PTF light curves, we find
populations of changing-look AGN: 8 galaxies that are variable in Stripe 82,
but quiescent in PTF, and 15 galaxies where the reverse is true. Our PTF work
demonstrates the promise of long-term optical variability searches in low-mass
galaxies for finding AGNs missed by other selection techniques.
