The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system "intersect" in finite time with probability one, is extended to cover situations when the coupling is unnecessarily Markovian and its components are only converging (in a certain sense) to each other with time. Under these assumptions the unique ergodicity of the original Markov process is proven. A price for this generalization is the weak convergence to the unique invariant measure instead of the strong one. Applying these ideas to infinite interacting particle systems we consider even more involved situations when the unique ergodicity can be proven only for a restriction of the original system to a certain class of initial distributions (e.g. translational invariant ones). Questions about the existence of invariant measures with a given particle density are discussed as well.1 One might argue that in this toy model all "states" of the phase space except for the origin are "unessential" and this is the main reason of such behavior. This is not the case and we shall return to this question in the analysis of infinite particle systems in Section 5.

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Formation of tissue-specific transcriptional programs underlies multicellular development, but how the chromatin landscape influences transcription is not fully understood. Here we comprehensively resolve differential transcriptional and chromatin states during Drosophila dorsoventral (DV) patterning. We find that RNA Polymerase II pausing is established at DV promoters prior to zygotic genome activation (ZGA), that pausing persists irrespective of cell fate, but that release into productive elongation is tightly regulated and accompanied by tissue-specific P-TEFb recruitment. DV enhancers acquire distinct tissue-specific chromatin states through CBP-mediated histone acetylation that predict the transcriptional output of target genes, whereas promoter states are more tissue invariant. Transcriptome-wide inference of burst kinetics in different cell types revealed that while DV genes are generally characterized by a high burst size, either burst size or frequency can differ between tissues. The data suggest that pausing is established by pioneer transcription factors prior to ZGA and that release from pausing is imparted by enhancer chromatin state to regulate bursting in a tissue-specific manner in the early embryo. Our results uncover how developmental patterning is orchestrated by tissue-specific bursts of transcription from Pol II primed promoters in response to enhancer regulatory cues.

We establish the Strong Poisson Hypothesis for symmetric closed networks. In particular, the asymptotic independence of the nodes as the size of the system tends to infinity is proved.

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