Ergodicity for the randomly forced Korteweg-de Vries-Burgers equation

2025/09/02 by Peng Gao, Gao, Peng
Economics, Econometrics and Finance · Engineering · Mathematics · #Advanced Mathematical Physics Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2509.01921

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Our goal in this paper is to investigate ergodicity of the randomly forced Korteweg-de Vries-Burgers(KdVB) equation driven by non-additive white noise. Under reasonable conditions, we show that exponential ergodicity for KdVB equation driven by a space-time localised noise and ergodicity for KdVB equation driven by a multiplicative white noise. Our proof is based on some newly developed analytical properties for KdVB equation, such as Carleman estimate, truncated observability inequality, Foiaş-Prodi estimate. Combining these analytical properties with coupling method and asymptotic coupling method, we can investigate the long time behavior of randomly forced KdVB equation.

Citations

• Uniqueness of the invariant measure and asymptotic stability for the 2D Navier Stokes equations with multiplicative noise
• On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation
• Mixing via controllability for randomly forced nonlinear dissipative PDEs
• Generalized couplings and ergodic rates for SPDEs and other Markov\n models
• Controllability implies mixing II. Convergence in the dual-Lipschitz metric
• Exponential mixing for a class of dissipative PDEs with bounded degenerate noise
• Generalized couplings and convergence of transition probabilities
• Exponential mixing for the white - forced damped nonlinear wave equation
• Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems
• Control and mixing for 2D Navier-Stokes equations with space-time localised noise
• Local and global well-posedness of the stochastic KdV-Burgers equation
• Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion

Related

• Exact Solutions of Stochastic Burgers-KdV Equation with variable coefficients
• Local and global well-posedness of the stochastic KdV-Burgers equation
• Global dynamics for the stochastic KdV equation with white noise as initial data
• Exponential mixing for the randomly forced NLS equation
• On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation