\label{sec:car}
The optical light curves of quasars are generated by fluctuations in the brightness of the accretion disk with structure in the time series on the order of days [REFS]. Since these fluctuations are coherent, the implication is that the size of the accretion disk is roughly \(R_{\rm src} \sim 10^{16}\) cm (which will be important for the microlensing calculation in §\ref{sec:microlensing}). These fluctuations have been shown to be well described by a Continuous Auto Regressive (CAR) process. First described by [REFS], the CAR process is a damped random walk and is equivalent to a Gaussian Process in which the covariance between two points on the light curve decreases as a function of their temporal separation. Using data from the MACHO survey [REF], [REF] fit a CAR process to (Greg: how many?) r-band MACHO quasar light curves. The CAR process is given by [see Appendix in REFS], \[M(t) = e^{-t/\tau} M(0) + \bar{M}(1-e^{-t/\tau}) + \sigma\int_{0}^{t} e^{-(t-s)/\tau} dB(s),\] where \(M\) is the magnitude of an image, \(\tau\) is a characteristic timescale in days, \(\bar{M}\) is the mean magnitude of the light curve in the absence of fluctuations, and \(\sigma\) is the characteristic amplitude of the fluctuations in mag/day\(^{1/2}\). In this model, fluctuations are generated by the integral term where \(dB(s)\) is a normally distributed value with mean zero and standard deviation \(dt\). By fitting the above model to the data, [REF] generated a distribution of \(\tau\) and \(\sigma\) for the MACHO quasars; we show typical examples of the CAR process with reasonable values for those parameters in Figure \ref{fig:example_lcs}.
(Greg: can you fill in the REFS please?)
While the damped random walk process provides a good description of the data obtained so far, it is not yet clear whether it will remain so for longer baseline, higher cadence, or multi-filter light curves. The different emission regions of an AGN (different parts of the accretion disk, broad and narrow line clouds, etc.) are likely to vary in different ways, suggesting that sums of stochastic processes could provide more accurate descriptions \citep{KellyMultipleDRWs}. These subcomponents would likely need parameters drawn from different distributions to the one above, and the correlations between the processes may need to be taken into account as well. Nevertheless, the success of the CAR model to date makes it a reasonable place to begin when simulating LSST-like AGN light curves.