This paper investigates the exponential convergence properties of Langevin diffusion and its discrete approximations. The authors study the Langevin diffusion, which is a continuous-time method for approximating a given distribution \(\pi\) using the stochastic differential equation \(d\mathbf{L}_t = d\mathbf{W}_t + \frac{1}{2} \nabla \log \pi(\mathbf{L}_t) dt\). They find conditions under which this diffusion converges exponentially quickly to \(\pi\), particularly for distributions with exponential tails of the form \(\pi(x) \propto \exp(-|\gamma| x^\beta)\), where \(\beta \geq 1\) ensures exponential convergence. The paper then examines the conditions under which discrete approximations to the Langevin diffusion converge. It is shown that even when the diffusion itself converges, naive discretizations may not. The authors introduce a "Metropolis-adjusted" version of the algorithm and find conditions for this to converge exponentially as well. However, they also demonstrate that even this modified version may not converge exponentially fast if the tails of the target density are lighter than exponential. Finally, the authors discuss a truncated form of the algorithm, which they argue should avoid the issues of other forms in practical applications. The paper provides detailed proofs and examples to support these findings, including the classification of the behavior of the Langevin diffusion for different classes of distributions and the conditions under which the unadjusted and Metropolis-adjusted Langevin algorithms converge geometrically.This paper investigates the exponential convergence properties of Langevin diffusion and its discrete approximations. The authors study the Langevin diffusion, which is a continuous-time method for approximating a given distribution \(\pi\) using the stochastic differential equation \(d\mathbf{L}_t = d\mathbf{W}_t + \frac{1}{2} \nabla \log \pi(\mathbf{L}_t) dt\). They find conditions under which this diffusion converges exponentially quickly to \(\pi\), particularly for distributions with exponential tails of the form \(\pi(x) \propto \exp(-|\gamma| x^\beta)\), where \(\beta \geq 1\) ensures exponential convergence. The paper then examines the conditions under which discrete approximations to the Langevin diffusion converge. It is shown that even when the diffusion itself converges, naive discretizations may not. The authors introduce a "Metropolis-adjusted" version of the algorithm and find conditions for this to converge exponentially as well. However, they also demonstrate that even this modified version may not converge exponentially fast if the tails of the target density are lighter than exponential. Finally, the authors discuss a truncated form of the algorithm, which they argue should avoid the issues of other forms in practical applications. The paper provides detailed proofs and examples to support these findings, including the classification of the behavior of the Langevin diffusion for different classes of distributions and the conditions under which the unadjusted and Metropolis-adjusted Langevin algorithms converge geometrically.