  Langevin MCMC gradient optimization is a class of increasingly popular
methods for estimating a posterior distribution. This paper addresses the
algorithm as applied in a decentralized setting, wherein data is distributed
across a network of agents which act to cooperatively solve the problem using
peer-to-peer gossip communication. We show, theoretically, results in 1) the
time-complexity to $\epsilon$-consensus for the continuous time stochastic
differential equation, 2) convergence rate in $L^2$ norm to consensus for the
discrete implementation as defined by the Euler-Maruyama discretization and 3)
convergence rate in the Wasserstein metric to the optimal stationary
distribution for the discretized dynamics.
