Langevin随机偏微分方程的弛豫与稳态熵产生：Dirichlet形式方法
Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach
AI总结：
本文用Dirichlet形式框架研究预条件Langevin随机偏微分方程的弛豫与稳态熵产生，给出熵耗散公式、熵产生率识别及显式衰减界。
AI中文摘要：
我们为预条件Langevin随机偏微分方程中的弛豫和稳态熵产生发展了一个Dirichlet形式框架。在无穷维空间中，基于环境Lebesgue密度和相应概率流密度的通常Fokker-Planck计算通常不可用。对于细致平衡类，Gibbs态空间上的拟正则对称Dirichlet形式给出了正则密度的精确de Bruijn熵耗散公式，以及对任意有限熵初始律的积分不等式。自伴性给出细致平衡和稳态路径反转，而坐标鞅准则将相关过程与给定的SPDE识别。对于一维$\Phi^4_1$和凸Allen-Cahn型Gibbs动力学，我们将已建立的强适定性理论与对数导数、形式闭包、拟正则性和形式-SPDE对应的直接验证相结合，并分别获得指数为$2$和$2(1-\lambda/m)$的相对熵衰减界，后者中$m>\lambda$。在远离细致平衡时，有界斜自伴线性强迫在不需要强迫与协方差可交换的情况下保持高斯不变律。我们识别了柱可观测量上的反对称作用及其Cameron-Martin流，并证明了平方流能量等于由前向-反向路径空间相对熵每单位时间定义的稳态熵产生率，以及Galerkin速率的单调极限。在可交换条件下，我们还获得了质量淬灭后的显式瞬态Onsager分解。排斥过程和Gaussian转子提供了有限状态和高斯基准。
英文摘要：
We develop a Dirichlet-form framework for relaxation and steady-state entropy production in preconditioned Langevin stochastic partial differential equations. In infinite dimensions, the usual Fokker--Planck calculations based on ambient Lebesgue densities and the corresponding probability-current densities are generally unavailable. For the detailed-balance class, a quasi-regular symmetric Dirichlet form on the Gibbs state space yields an exact de Bruijn entropy-dissipation formula for regular densities and an integrated inequality for arbitrary finite-entropy initial laws. Self-adjointness gives detailed balance and stationary path reversal, while a coordinate-martingale criterion identifies the associated process with the prescribed SPDE. For the one-dimensional $Φ^4_1$ and convex Allen--Cahn-type Gibbs dynamics, we combine the established strong well-posedness theory with direct verification of the logarithmic derivatives, form closure, quasi-regularity and form--SPDE correspondence, and obtain relative-entropy decay bounds with exponents $2$ and $2(1-λ/m)$, respectively, with $m>λ$ in the latter case. Away from detailed balance, bounded skew-adjoint linear forcing preserves a Gaussian invariant law without requiring commutation between the forcing and covariance. We identify the antisymmetric action on cylinder observables and its Cameron--Martin current and prove that the squared current energy equals the steady-state entropy-production rate defined by forward--reverse path-space relative entropy per unit time, as well as the monotone limit of the Galerkin rates. Under commutation, we additionally obtain an explicit transient Onsager decomposition after a mass quench. Exclusion processes and Gaussian rotors provide finite-state and Gaussian benchmarks.
