We provide CoRoT and Spitzer light curves, as well as broad-band multi-wavelength photometry and high resolution, multi- and single-epoch spectroscopy for 17 classical T Tauris in NGC 2264 whose CoRoT light curves (LCs) exemplify the stochastic LC class as defined in Cody et al. (2014). The most probable physical mechanism to explain the optical variability in this LC class is time-dependent mass accretion onto the stellar photosphere, producing transient hot spots. As evidence in favor of this hypothesis, multi-epoch high resolution spectra for a subset of these stars shows that their veiling levels also vary in time and that this veiling variability is consistent in both amplitude and timescale with the optical LC morphology. Furthermore, the veiling variability is well-correlated with the strength of the HeI 6678A emission line, a feature predicted by models to arise in accretion shocks on or near the stellar photosphere. Stars with accretion burst LC morphology (Stauffer et al. 2014) are also attributed to variable mass accretion. Both the stochastic and accretion burst LCs can be explained by a simple model of randomly occurring flux bursts, with the stochastic LC class having a higher frequency of lower amplitude events. Based on their UV excesses, veiling, and mean Ha equivalent widths, members of the stochastic LC class have only moderate time-averaged mass accretion rates. The most common feature of their Ha profiles is for them to exhibit blue-shifted absorption features, most likely originating in a disk wind. The lack of periodic signatures in the LCs suggests that little of the variability is due to long-lived hot spots rotating into or out of our line of sight; instead, the primary driver of the observed photometric variability is likely to be instabilities in the inner disk that lead to variable mass accretion.
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