Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach
Abstract
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 Φ41 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.
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BibTeX
@misc{fan2026relaxation,
title = {{Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach}},
author = {Shuyuan Fan and Yuanke Chen and Lifei Wang and Jinqiao Duan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18201},
eprint = {2609.18201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18201},
}
RIS
TY - UNPB
TI - Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach
AU - Shuyuan Fan
AU - Yuanke Chen
AU - Lifei Wang
AU - Jinqiao Duan
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.18201
UR - https://arxiv.org/abs/2609.18201
AB - 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.
ER -