<Invariant distribution of an Itô diffusion>, specialized to <Underdamped Langevin dynamics>, has density
$$
\pi(x,p)=Z^{-1}\exp\left\{-U(x)-\frac{\lVert p\rVert^2}{2\eta}\right\}.
$$
Thus $X$ 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
$$
\begin{aligned}
P_{k+1}
&=P_k-\gamma\delta P_k-\eta\delta\nabla U(X_k)
+\sqrt{2\gamma\eta\delta}\,\xi_k,\\
X_{k+1}&=X_k+\delta P_k.
\end{aligned}
$$

Solution

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

{c}
{wiki=Euler–Maruyama_method}

For $dX_t=a(X_t)dt+\sigma(X_t)dW_t$, the Euler-Maruyama method advances
$$
X_{n+1}=X_n+a(X_n)\Delta t+\sigma(X_n)\Delta W_n,
$$
where $\Delta W_n\sim N(0,\Delta t)$.


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