Existence and polynomial dependence of uniform derivative bounds for transition densities (even in Euclidean space)

Determine whether the uniform bounds on derivatives of transition densities along sample paths, assumed in Assumption 3 of Wang, Lei, and Panageas (2020) to analyze the geometric Euler–Maruyama discretization of Langevin dynamics, actually exist even in Euclidean space; and, if such bounds exist, ascertain whether they can be made to depend only polynomially on the dimension and other problem parameters.

Background

In discussing prior analyses of the geometric Euler–Maruyama discretization, the paper notes that Wang et al. (2020) rely on uniform bounds on certain derivatives of transition densities along the sample path (their Assumption 3). The authors point out that the validity of such bounds is unclear even in the simpler Euclidean setting, and that the dependence of any such bounds on dimension may be unfavorable.

This uncertainty motivates the paper’s alternative approach that avoids requiring such uniform derivative bounds. Nonetheless, the status of the existence and the dimensional dependence of these bounds remains an explicit unresolved question highlighted by the authors.