It\^o maps for any-step SDEs

Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the It\^o map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The It\^o map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, It\^o maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as a useful primitive for posterior sampling and stochastic control.