Kinetic Brownian motion on Riemannian manifolds

Jürgen Angst (1) , Ismaël Bailleul (1) , Camille Tardif (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 UPMC - Université Pierre et Marie Curie - Paris 6

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle T 1 M of a Riemannian manifold (M, g), collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter σ quantifying the size of the noise. Projection on M of these processes provides random C 1 paths in M. We show, both qualitively and quantitatively, that the laws of these M-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter σ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when σ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.

Mots clés: Diffusion processes; finite speed propagation; Riemannian manifolds; homogeniza-tion; rough paths theory; Poisson boundary

Domaines: Mathématiques [math]

Jürgen Angst, Ismaël Bailleul, Camille Tardif. Kinetic Brownian motion on Riemannian manifolds. Electronic Journal of Probability, 2015, 20 (none), pp.110. ⟨10.1214/EJP.v20-4054⟩. ⟨hal-01263340⟩