Kinetic Brownian motion on Riemannian manifolds

Abstract

International audienceWe 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

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

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Homogenisation for anisotropic kinetic random motions

“…The presence of anisotropy drastically complexifies the approach and computations compared to the isotropic framework. Namely, in the isotropic Euclidean setting considered in Section 2.2 of [ABT15] and which is the core of the proof when associated with rough paths techniques, the homogenisation of kinetic Brownian motion was proved using Itô calculus and standard martingale techniques. As it will be clear in Section 2 below, the Doob-Meyer decomposition of the velocity process given by equation (1.1) gets more involved here, its invariant measure is not likely to be easy to describe, and martingale techniques need explicit solutions of the Poisson equation which seems hopeless in this context.…”

Entropy Anomaly in Langevin–Kramers Dynamics with a Temperature Gradient, Matrix Drag, and Magnetic Field

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