Electronic Journal of Probability

Kinetic Brownian motion on Riemannian manifolds

Jürgen Angst, Ismaël Bailleul, and Camille Tardif

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 110, 40 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Diffusion processes finite speed propagation Riemannian manifolds homogenization rough paths theory Poisson boundary

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Angst, Jürgen; Bailleul, Ismaël; Tardif, Camille. Kinetic Brownian motion on Riemannian manifolds. Electron. J. Probab. 20 (2015), paper no. 110, 40 pp. doi:10.1214/EJP.v20-4054. https://projecteuclid.org/euclid.ejp/1465067216

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