Fluctuation analysis is an experimental design in molecular biology where fluctuations across independent clones grown from genetically identical single cells allow to draw inferences about heritable mechanisms. Originally invented by Luria and Delbrück in the 1940s to demonstrate that in bacteria, genetic mutations arise in the absence of selective pressure, fluctuation analysis has been used more recently in mammalian cells to demonstrate the existence of transcriptional memory, that is, slow fluctuations in gene expression that persist for multiple cell divisions and are associated with biologically distinct behaviors. I will present ongoing work on a mathematical model for fluctuation analysis of gene expression using stochastic processes on trees in which genes exhibit transcriptional memory when their half-life exceeds twice the doubling time of cell proliferation. In equilibrium, kinetic parameters and causal links between genes can be reconstructed from correlations across independent clones, a surprising example of a fluctuation-dissipation relation that does not require two-time correlations. I will conclude by speculating on a potential role of transcriptional memory in establishing long-range spatial correlations in gene expression that have been observed in spatial transcriptomics data.