  In this paper, we present conditions for identifying the generator of a
linear stochastic differential equation (SDE) from the distribution of its
solution process with a given fixed initial state. These identifiability
conditions are crucial in causal inference using linear SDEs as they enable the
identification of the post-intervention distributions from its observational
distribution. Specifically, we derive a sufficient and necessary condition for
identifying the generator of linear SDEs with additive noise, as well as a
sufficient condition for identifying the generator of linear SDEs with
multiplicative noise. We show that the conditions derived for both types of
SDEs are generic. Moreover, we offer geometric interpretations of the derived
identifiability conditions to enhance their understanding. To validate our
theoretical results, we perform a series of simulations, which support and
substantiate the established findings.
