We provide a complete characterisation of the class of one-dimensional time-homogeneous diffusions consistent with a given law at an exponentially distributed time using classical results in diffusion theory. To illustrate we characterise the class of diffusions with the same distribution as Brownian motion at an exponentially distributed time.
"The Wronskian parametrises the class of diffusions with a given distribution at a random time." Electron. Commun. Probab. 17 1 - 8, 2012. https://doi.org/10.1214/ECP.v17-1976