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.
Highlights
1.1 Motivations and related worksWe introduce in this work a one parameter family of diffusion processes that model physical phenomena with a finite speed of propagation, collectively called kinetic Brownian motion
We show that kinetic Brownian motion interpolates between geodesic and Brownian motions, as σ ranges from 0 to ∞, leading to a kind of homogenization
Kinetic Brownian motion is the Riemannian analogue of a class of diffusion processes on Lorentzian manifolds that was introduced by Franchi and Le Jan in [FLJ07], as a generalization to a curved setting of a process introduced by Dudley [Dud66] in Minkowski spacetime
Summary
We introduce in this work a one parameter family of diffusion processes that model physical phenomena with a finite speed of propagation, collectively called kinetic Brownian motion. Kinetic Brownian motion is the Riemannian analogue of a class of diffusion processes on Lorentzian manifolds that was introduced by Franchi and Le Jan in [FLJ07], as a generalization to a curved setting of a process introduced by Dudley [Dud66] in Minkowski spacetime These processes model the motion in spacetime of a massive object subject to Brownian fluctuations of its velocity. This is the main content of Theorem 2.2, which is proved using rough paths theory. Denoting by Γkij the Christoffel symbol of the Levi-Civita connection associated with the above coordinates, the vector fields Vi and H1 have the following expressions in these local coordinates
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
