The goal is to efficiently sample from a target distribution $\pi \propto e^{-f(x)}$, in high dimensional or non-convex settings, direct sampling difficult and usually we define a stochastic process whose stationary distribution is $\pi(x)$. For the purposes of this report we will restrict ourselves to sampling approaches based on Langevin diffusion, which is a stochastic process $(X_t){t \geq 0}$, where $X_t \in \mathbb{R}^d$ evolves according to the stochastic differential equation: $$ \mathrm{d}X_t = -\nabla f(X_t) \mathrm{d}t + \sqrt{2} \mathrm{d}B_t, $$ where $(B_t){t \geq 0}$ is the standard Brownian motion in $\mathbb{R}^d$.
When $f(x)$ is suitably smooth, it can be shown that $X_t$ has $\pi(x)$ as a stationary distribution. Additionally, the transition kernel $P_{\mathrm{L}}^t(x, A)$ defined as:
$$ P_{\mathrm{L}}^t(x, A)=P\left(X_t \in A \mid X_0=x\right), \quad t \geq 0 $$
satisfies the convergence property:
$$ \left|P_{\mathrm{L}}^t(x, \cdot)-\pi\right| \rightarrow 0 \quad \text { as } t \rightarrow \infty $$
for all $x \in \mathbb{R}^k$. Here, $|\cdot|$ denotes the total variation norm, and the convergence indicates that the distribution of $X_t$ approaches the target distribution $\pi(x)$ over time.