  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.
