  Inferring the causal structure underlying stochastic dynamical systems from
observational data holds great promise in domains ranging from science and
health to finance. Such processes can often be accurately modeled via
stochastic differential equations (SDEs), which naturally imply causal
relationships via "which variables enter the differential of which other
variables". In this paper, we develop a kernel-based test of conditional
independence (CI) on "path-space" -- e.g., solutions to SDEs, but applicable
beyond that -- by leveraging recent advances in signature kernels. We
demonstrate strictly superior performance of our proposed CI test compared to
existing approaches on path-space and provide theoretical consistency results.
Then, we develop constraint-based causal discovery algorithms for acyclic
stochastic dynamical systems (allowing for self-loops) that leverage temporal
information to recover the entire directed acyclic graph. Assuming faithfulness
and a CI oracle, we show that our algorithms are sound and complete. We
empirically verify that our developed CI test in conjunction with the causal
discovery algorithms outperform baselines across a range of settings.
