Plug-and-Play is a Reverse SDE: A Unified Convergence Theory for Annealed Denoising Priors

Plug-and-Play (PnP) image reconstruction replaces the proximal step of an inverse-problem optimizer by an off-the-shelf denoiser, and has become the de facto standard when the prior is a deep denoiser or a diffusion score. Yet its convergence theory has lagged practice: classical fixed-point analyses require the denoiser to be non-expansive or Lipschitz, assumptions that modern annealed priors violate by construction. We show that the PnP iteration $v^{t+1}=D(h(v^t;y),{\sigma }_t)$ is, under a Gaussian-denoising (Tweedie) assumption, the Euler–Maruyama discretization of a time-inhomogeneous It\^o SDE whose drift is the sum of the data-fidelity step and the score of the noise-perturbed prior. When the denoiser is the MMSE denoiser and $h$ is a posterior-gradient step, this SDE is precisely the deterministic-drift reverse SDE of variance-exploding diffusion models. Building on this identification, we give a complete convergence chain: (i) a global-Lipschitz EM discretization error bound; (ii) geometric ergodicity of the limiting homogeneous SDE under dissipativity and a strictly positive terminal noise level ${\sigma }_{\min }>0$; and (iii) weak convergence of the PnP iterates to the invariant law $\pi$, with a Wasserstein-$2$ bound $W_2(\pi ,p_{{\sigma }_{\min }}(x\mid y))\le C{\epsilon }_{\mathrm{score}}/m$ that quantifies how score-approximation error controls the posterior bias. Boundedness of the denoiser alone is not sufficient: the guarantee is dissipativity $+$ a non-vanishing noise floor $+$ bounded drift. Experiments on real DRUNet and a pretrained ADM diffusion prior, across denoising, inpainting, super-resolution, and medical imaging, corroborate the EM rate, the ${\sigma }_{\min }$ threshold, and the posterior-distance bound.