Multidimensional moment-bound conjecture for PDMP approximations

Let $\psi$ satisfy the growth bounds in $$, and let $H(x,v)$ be the function in inequality $$. The mesh satisfies $0=t_0\leq t_1\leq\cdots$ with $\delta_k=t_k-t_{k-1}<\delta_0$, and the approximations are generated by Algorithms $$ and $$. Multidimensional moment-bound conjecture. Inequality $$ holds for both Algorithms $$ and $$. The claim is proved in the one-dimensional case; its extension to general dimension is left open and is supported by numerical experiments.