Stable maintenance of gene regulatory programs is essential for normal function in multicellular organisms. Epigenetic mechanisms, and DNA methylation in particular, are hypothesized to facilitate such maintenance by creating cellular memory(1-4) that can be written during embryonic development(5,6) and then guide cell-type-specific gene expression(7). Here we develop new methods for quantitative inference of DNA methylation turnover rates, and show that human embryonic stem cells preserve their epigenetic state by balancing antagonistic processes that add and remove methylation marks rather than by copying epigenetic information from mother to daughter cells. In contrast, somatic cells transmit considerable epigenetic information to progenies. Paradoxically, the persistence of the somatic epigenome makes it more vulnerable to noise, since random epimutations can accumulate to massively perturb the epigenomic ground state. The rate of epigenetic perturbation depends on the genomic context, and, in particular, DNA methylation loss is coupled to late DNA replication dynamics. Epigenetic perturbation is not observed in the pluripotent state, because the rapid turnover-based equilibrium continuously reinforces the canonical state. This dynamic epigenetic equilibrium also explains how the epigenome can be reprogrammed quickly(8) and to near perfection(9) after induced pluripotency.