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 $Φ^4_1$ 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.