[论文解读] Solving Poisson's equation for Wasserstein contractive Markov chains

Julian Hofstadler|arXiv (Cornell University)|Feb 22, 2026

Geometric Analysis and Curvature Flows被引用 0

一句话总结

该论文在 Wasserstein 收缩下研究一般状态空间马尔可夫链的 Poisson 方程，证明 Lipschitz 与某些 L^p 力 forcing 函数下解的存在性与正则性，并导出极大不等式。

ABSTRACT

We study Poisson's equation in the context of general state space Markov chains. For chains satisfying a contraction assumption w.r.t. a Wasserstein distance, we show that a solution exists for Lipschitz functions and investigate its regularity properties. If the kernel is additionally reversible we are also able to show that solutions for $L^p$ functions exist. Combining our findings with Doob's inequalities for martingales we derive maximal inequalities for contractive Markov chains. A number of examples is provided to demonstrate the applicability of our results, in particular in the context of Markov chain Monte Carlo methods.

研究动机与目标

• Motivate and analyze Poisson’s equation for general state space Markov chains.
• Establish existence and regularity of solutions under Wasserstein contraction.
• Extend results to L^p settings under reversibility and derive integrability bounds.
• Provide examples including MCMC methods to demonstrate applicability.

提出的方法

• Use spectral properties of Markov operators on Banach spaces of centred Lipschitz functions.
• Represent solutions via Neumann series (Id - P)^{-1} on the appropriate function space.
• Employ Kantorovich-Rubinstein duality to relate operator norms to Wasserstein contraction (tau).
• Prove existence of unique solutions u_f in L_0^d for Lipschitz forcing f, with bounds ||u||_d ≤ Λ||f||_d.
• Derive L^p bounds for u under additional moment conditions and reversibility.

实验结果

研究问题

• RQ1 Under Wasserstein contractivity, does Poisson’s equation have a unique solution in the centred Lipschitz function space for a given forcing f?
• RQ2 What regularity (L^p bounds) can be established for Poisson solutions, and how does reversibility affect the L^p theory?
• RQ3 Can maximal inequalities for the partial sums S_n f be derived from Poisson’s equation in this setting?
• RQ4 How do these results apply to MCMC algorithms and other contractive Markov chains?
• RQ5 What are explicit bounds in terms of Wasserstein contraction constants and eccentricity measures?

主要发现

• There exists a unique solution u_f to Poisson’s equation in the centred Lipschitz space for each f with finite p-eccentricity, with ||u||_d ≤ Λ||f||_d.
• If E_p(x0) < ∞ for some x0 and p ≥ 1, then π(|f|^p) and π(f) are well-defined for Lipschitz f, enabling Poisson’s equation to be well-posed.
• Under additional integrability of E_1, stronger L^p bounds for u can be obtained, including explicit L^{p0} bounds.
• If the chain is reversible, L^p existence results for Poisson’s equation extend, broadening previous uniformly ergodic cases.
• The results yield novel maximal inequalities for contractive Markov chains, enabling path-wise convergence analysis for algorithms like No-U-Turn-Sampler (NUTS).
• The framework includes examples relevant to MCMC (e.g., Heat bath, Slice Sampling, MALA, independent Metropolis-Hastings) showing contractivity and applicability.