Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach

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 ${\mathrm{\Phi }}_1^4$ 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 $2(1-\lambda \mathrm{/}m)$, respectively, with $m>\lambda$ 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.