Causal transport on path space

We study properties of causal couplings for probability measures on the space of continuous functions. We first provide a characterization of bicausal couplings between weak solutions of stochastic differential equations. We then provide a complete description of all such bicausal Monge couplings. In particular, we show that bicausal Monge couplings of $d$-dimensional Wiener measures are induced by stochastic integrals of rotation-valued integrands. As an application, we give necessary and sufficient conditions for bicausal couplings to be induced by Monge maps and show that such bicausal Monge transports are dense in the set of bicausal couplings between laws of SDEs with regular coefficients.