Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 4 / c

Invariant distribution of an Itô diffusion, specialized to Underdamped Langevin dynamics, has densityThus $X$ has density proportional to $e^{-U(x)}$, and conditionally and marginally $P\sim N(0,\eta I_d)$. The Hamiltonian transport between $x$ and $p$ preserves this density, while the Ornstein-Uhlenbeck process in momentum has exactly that Gaussian invariant law.

The Euler-Maruyama method with step size $\delta$ and independent ${\xi }_k\sim N(0,I_d)$ is