Variational and Bayesian methods are two approaches that have been widely used to solve image reconstruction problems. In this paper, we propose original connections between Hamilton--Jacobi (HJ) partial differential equations and a broad class of Bayesian methods and posterior mean estimators with Gaussian data fidelity term and log-concave prior. Whereas solutions to certain first-order HJ PDEs with initial data describe maximum a posteriori estimators in a Bayesian setting, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. These connections allow us to establish several representation formulas and optimal bounds involving the posterior mean estimate. In particular, we use these connections to HJ PDEs to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of twice continuously differentiable functions, and furthermore we derive a representation formula for these functions.