Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

2025/11/15 by Shan Huang, Xiaoyue Li, Huang, Shan +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2511.12124

openalex publication_date 2025/11/15 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the L^p-Wasserstein distance (p\geqslant 1). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time 1/2-order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a 1/2-order convergence rate for the invariant measures of the TEM scheme and the exact solution in the L¹-Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

Citations

• Non-asymptotic Error Analysis of Explicit Modified Euler Methods for Superlinear and Non-contractive SODEs
• Geometric ergodicity of modified Euler schemes for SDEs with super-linearity
• Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs
• L²-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications
• Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients
• Convergence of kinetic Langevin samplers for non-convex potentials
• Uniform-in-time estimates for mean-field type SDEs and applications
• On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients
• L²-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems
• Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)
• Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting
• The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients
• Strict Kantorovich contractions for Markov chains and Euler schemes with general noise
• Approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations
• Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations
• Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity
• Non-asymptotic bounds for sampling algorithms without log-concavity
• Quantitative contraction rates for Markov chains on general state spaces
• The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching
• The Tamed Unadjusted Langevin Algorithm
• Approximation of Invariant Measure for Damped Stochastic Nonlinear Schrödinger Equation via an Ergodic Numerical Scheme
• Exponential Convergence in L^p-Wasserstein Distance for Diffusion Processes without Uniformly Dissipative Drift
• Approximation of the invariant measure with an Euler scheme for\n Stochastic PDE's driven by Space-Time White Noise
• Yet another look at Harris' ergodic theorem for Markov chains
• Explicit numerical approximations for McKean-Vlasov stochastic differential equations in finite and infinite time

Related

• Explicit approximation of the invariant measure for SDDEs with the nonlinear diffusion term
• Approximation of Invariant Measure for Damped Stochastic Nonlinear Schrödinger Equation via an Ergodic Numerical Scheme
• Approximation of the invariant measure with an Euler scheme for\n Stochastic PDE's driven by Space-Time White Noise
• Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme
• Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme