In discrete time, we usually want to use a step size $\eta$ to construct an iterative update rule for the discretization.
Unadjusted Langevin Algorithm (ULA)
Unadjusted Langevin Algorithm (ULA) is a simple discretization of the Langevin diffusion: $$ X_{k+1} = X_k - \eta \nabla f(X_k) + \sqrt{2\eta} \zeta_k, $$ where $\eta > 0$ is the step size and $\zeta_k \sim \mathcal{N}(0, I)$ are independent Gaussian random variables. The drift term $\eta \nabla f(X_k)$ introduces bias; for any fixed $\eta > 0$, the law $\rho_k$ of $X_k$ converges to a limiting distribution $\nu_\eta$, which differs from the true target $\nu$.
While, ULA can approximate sampling using a small stepsize to keep the bias within acceptable thresholds (\cite{dalalyan2017}, and \cite{durmus_non-asymptotic_2016} it is asymptotically biased due to its discretization structure being forward-flow and not the true gradient descent in the space of measures.
Underdamped Langevin Monte Carlo
ULMC incorporates momentum to address ULA’s slow mixing, allowing the particle to retain velocity across steps. The second order SDE is defined as: $$ \begin{aligned} d X_t & =V_t dt\ d V_t & =-\gamma V_t dt- \nabla f(X_t) d t+(\sqrt{2 \gamma}) d B_t \end{aligned} $$ where $\gamma \ge 0$ is the friction coefficient and $V_t$ represents velocity.