We consider noninteracting infinite particles, each of which follows a diffusion with generator $L \equiv (D^2 + D)/2$. The presence of many invariant distributions makes the situation radically different from the more familiar case where strong ergodicity assumptions are made. Explicit large deviation rates for the empirical density are obtained. The dependence of the rates on the initial distribution is strong and can be seen clearly. Some variational formulas for the scattering data associated with $L$ are also obtained.
"Large Deviations for Noninteracting Infinite Particle Systems." Ann. Probab. 16 (4) 1537 - 1558, October, 1988. https://doi.org/10.1214/aop/1176991582