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2015 Kinetic Brownian motion on Riemannian manifolds
Jürgen Angst, Ismaël Bailleul, Camille Tardif
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Electron. J. Probab. 20: 1-40 (2015). DOI: 10.1214/EJP.v20-4054

Abstract

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $\sigma$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ 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 $\sigma$ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when $\sigma$ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.

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Jürgen Angst. Ismaël Bailleul. Camille Tardif. "Kinetic Brownian motion on Riemannian manifolds." Electron. J. Probab. 20 1 - 40, 2015. https://doi.org/10.1214/EJP.v20-4054

Information

Accepted: 19 October 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1329.60274
MathSciNet: MR3418542
Digital Object Identifier: 10.1214/EJP.v20-4054

Subjects:
Primary: 60J60
Secondary: 58J65, 60J45

JOURNAL ARTICLE
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