Direct first-order comparison for the discretized underdamped sampler

Establish a first-order comparison between the discretized critically damped underdamped reverse sampler and the exact reverse process itself, controlling the sampler’s full relative Fisher divergence with respect to the exact reverse marginals rather than only with respect to the twin chain generated from the exact initialization through the same numerical kernels.

Background

Theorem 5.1 analyzes the discretized critically damped underdamped reverse process indirectly. It compares the numerical sampler with a twin chain that begins from the exact terminal initialization and is propagated through the same frozen-score numerical kernels. This construction isolates initialization error and permits the use of a moving hypocoercive metric, but it does not directly compare the numerical sampler with the exact reverse marginals.

The paper notes that a path-space Girsanov estimate gives a discretization bound between the exact reverse process and the twin chain, but that this estimate does not automatically yield a pointwise or time-averaged full relative-Fisher bound against the exact reverse marginals. Closing this transfer gap would provide the direct first-order discretization guarantee suggested by the paper’s theory.