Heavy-tailed controlled diffusions in the underdamped regime

Investigate the controlled diffusion framework for sampling when the forward noising process is a Student’s t-based heavy-tailed underdamped diffusion defined by dX_t = M^{-1} V_t dt and dV_t = -((α + d)/α)·X_t/(1 + ||X_t||^2/α) dt − Γ M^{-1} V_t dt + √(2Γ) dB_t. In particular, derive tractable reverse-time dynamics with appropriate control terms and develop stable numerical integrators for this regime where linear SDE techniques used for Ornstein–Uhlenbeck noising do not apply.

Background

The paper introduces MultCDiff, a multiscale controlled diffusion sampler that leverages explicit reverse-time dynamics available for an Ornstein–Uhlenbeck (OU) noising process, including carefully designed control terms and a symmetric splitting integrator. This construction critically relies on linear SDE structure in the forward dynamics, enabling closed-form expressions for the forward marginals and their conditional scores.

The authors propose extending controlled diffusions to heavy-tailed settings using a Student’s t noising process in the underdamped regime. However, the resulting forward dynamics have a non-linear drift in the velocity equation, so linear SDE techniques do not yield analytic forms. Consequently, the explicit derivations and constructions that underpin MultCDiff cannot be directly transferred, and a full treatment of reverse-time dynamics and control terms for this heavy-tailed case is deferred.