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We develop a space-time large-deviation point of view on Gibbs-non-Gibbstransitions in spin systems subject to a stochastic spin-flip dynamics. Usingthe general theory for large deviations of functionals of Markov processesoutlined in Feng and Kurtz [11], we show that the trajectory under thespin-flip dynamics of the empirical measure of the spins in a large block inZ^d satisfies a large deviation principle in the limit as the block size tendsto infinity. The associated rate function can be computed as the actionfunctional of a Lagrangian that is the Legendre transform of a certainnon-linear generator, playing a role analogous to the moment-generatingfunction in the Gartner-Ellis theorem of large deviation theory when this isapplied to finite-dimensional Markov processes. This rate function is used todefine the notion of "bad empirical measures", which are the discontinuitypoints of the optimal trajectories (i.e., the trajectories minimizing the ratefunction) given the empirical measure at the end of the trajectory. Thedynamical Gibbs-non-Gibbs transitions are linked to the occurrence of badempirical measures: for short times no bad empirical measures occur, while forintermediate and large times bad empirical mea- sures are possible. A futureresearch program is proposed to classify the various possible scenarios behindthis crossover, which we refer to as a "nature-versus-nurture" transition.

A large-deviation view on dynamical Gibbs-non-Gibbs transitions