Dynamical properties and characterization of gradient drift diffusions

Sébastien Darses (1) , Ivan Nourdin (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We study the dynamical properties of the Brownian diffusions having $\sigma\,{\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We characterize this class through the equality $D^2_+=D^2_-$, where $D_{+}$ (resp. $D_-$) denotes the forward (resp. backward) stochastic derivative of Nelson's type. Our proof is based on a remarkable identity for $D_+^2-D_-^2$ 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.

Mots clés: Gradient drift diffusion; Time reversal; Nelson stochastic derivatives; Kolmogorov theorem; Reversible diffusion; Stationary diffusion; Martingale problem

Domaines: Probabilités [math.PR]

Sébastien Darses, Ivan Nourdin. Dynamical properties and characterization of gradient drift diffusions. 2006. ⟨hal-00120426⟩