Dynamical properties and characterization of gradient drift diffusions

We study the dynamical properties of the Brownian diffusions having σ Id as diffusion coefficient matrix and b = ∇U as drift vector. We characterize this class through the equality D2 + = D 2 −, where D+ (resp. D−) denotes the forward (resp. backward) stochastic derivative of Nelson’s type. Our proof is based on a remarkable identity for D2 + −D2 − and on the use of the martingale problem. We also give a new formulation of a famous theorem of Kolmogorov concerning reversible diffusions. We finally relate our characterization to some questions about the complex stochastic embedding of the Newton equation which initially motivated of this work.