Efficient sampling from complex and high dimensional target distributions turns out to be a fundamental task in diverse disciplines such as scientific computing, statistics and machine learning. In this paper, we revisit the randomized Langevin Monte Carlo (RLMC) for sampling from high dimensional distributions without log-concavity. Under the gradient Lipschitz condition and the log-Sobolev inequality, we prove a uniform-in-time error bound in $\mathcal{W}_2$-distance of order $O(\sqrt{d}h)$ for the RLMC sampling algorithm, which matches the best one in the literature under the log-concavity condition. Moreover, when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, modified RLMC algorithms are proposed and analyzed, with non-asymptotic error bounds established. To the best of our knowledge, the modified RLMC algorithms and their non-asymptotic error bounds are new in the non-globally Lipschitz setting.

Standard first-order Langevin algorithms such as the unadjusted Langevin algorithm (ULA) are obtained by discretizing the Langevin diffusion and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they face two key limitations: (i) they require differentiable log-densities, excluding targets with non-differentiable components; and (ii) they generally fail to sample heavy-tailed targets. We propose anchored Langevin dynamics, a unified approach that accommodates non-differentiable targets and certain classes of heavy-tailed distributions. The method replaces the original potential with a smooth reference potential and modifies the Langevin diffusion via multiplicative scaling. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and provide an equivalent formulation derived via a random time change of the Langevin diffusion. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.

In this paper, we examine the problem of sampling from log-concave distributions with (possibly) superlinear gradient growth under kinetic (underdamped) Langevin algorithms. Using a carefully tailored taming scheme, we propose two novel discretizations of the kinetic Langevin SDE, and we show that they are both contractive and satisfy a log-Sobolev inequality. Building on this, we establish a series of non-asymptotic bounds in $2$-Wasserstein distance between the law reached by each algorithm and the underlying target measure.

The overdamped Langevin stochastic differential equation (SDE) is a classical physical model used for chemical, genetic, and hydrological dynamics. In this work, we prove that the drift and diffusion terms of a Langevin SDE are jointly identifiable from temporal marginal distributions if and only if the process is observed out of equilibrium. This complete characterization of structural identifiability removes the long-standing assumption that the diffusion must be known to identify the drift. We then complement our theory with experiments in the finite sample setting and study the practical identifiability of the drift and diffusion, in order to propose heuristics for optimal data collection.

In this article, we study the problem of sampling from distributions whose densities are not necessarily smooth nor log-concave. We propose a simple Langevin-based algorithm that does not rely on popular but computationally challenging techniques, such as the Moreau Yosida envelope or Gaussian smoothing. We derive non-asymptotic guarantees for the convergence of the algorithm to the target distribution in Wasserstein distances. Non asymptotic bounds are also provided for the performance of the algorithm as an optimizer, specifically for the solution of associated excess risk optimization problems.

In this article we consider sampling from log concave distributions in Hamiltonian setting, without assuming that the objective gradient is globally Lipschitz. We propose two algorithms based on monotone polygonal (tamed) Euler schemes, to sample from a target measure, and provide non-asymptotic 2-Wasserstein distance bounds between the law of the process of each algorithm and the target measure. Finally, we apply these results to bound the excess risk optimization error of the associated optimization problem.

This problem arises in many cases in machine learning, most notably in large-scale (mini-batch) Bayesian inference (Welling and Teh, 2011, Ahn et al., 2012) and nonconvex stochastic optimization (Raginsky et al., 2017). For the setting of Bayesian inference, one is interested in sampling from a posterior probability measure where U corresponds to the sum of the log-likelihood and the log-prior. For the nonconvex optimization, U(·) is the nonconvex cost function to be minimized. For large values ofβ, a sample from the target measure (1) is an approximate minimizer of the potential U (Raginsky et al., 2017). Consequently, nonasymptotic error bounds for the schemes, which are designed to sample from (1), can be used to obtain guarantees for Bayesian inference or nonconvex optimization. Sampling from a measure of the form (1) is also central in statistical physics (Binder et al., 1993), most notably in molecular dynamics Haile (1992).

In this paper an alternative approach to solve uncertain Stochastic Differential Equation (SDE) is proposed. This uncertainty occurs due to the involved parameters in system and these are considered as Triangular Fuzzy Numbers (TFN). Here the proposed fuzzy arithmetic in [2] is used as a tool to handle Fuzzy Stochastic Differential Equation (FSDE). In particular, a system of Ito stochastic differential equations is analysed with fuzzy parameters. Further exact and Euler Maruyama approximation methods with fuzzy values are demonstrated and solved some standard SDE.