Kinetic Brownian motion on Riemannian manifolds

2015/01/15 by Angst, Jürgen, Bailleul, Ismaël, Tardif, Camille
#58J65 #60J45 #60J60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1501.03679

Abstract

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle T¹ \mathcal M of a Riemannian manifold (\mathcal 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 \mathcal M of these processes provides random C¹ paths in \mathcal M. We show, both qualitively and quantitatively, that the laws of these \mathcal 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.

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