In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman N-coalescent back from time t consider the associated processes of total tree length as t increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as N tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).
"The Kingman tree length process has infinite quadratic variation." Electron. Commun. Probab. 19 1 - 12, 2014. https://doi.org/10.1214/ECP.v19-3318