We introduce a class of kinetic and anisotropic random motions px σ t , v σ t qtě0 on the unit tangent bundle T 1 M of a general Riemannian manifold pM, gq, where σ is a positive parameter quantifying the amount of noise affecting the dynamics. As the latter goes to infinity, we then show that the time rescaled process px σ σ 2 t qtě0 converges in law to an explicit anisotropic Brownian motion on M. Our approach is essentially based on the strong mixing properties of the underlying velocity process and on rough paths techniques, allowing us to reduce the general case to its Euclidean analogue. Using these methods, we are able to recover a range of classical results.
We study the spectrum of the kinetic Brownian motion in the space of d × d Hermitian matrices, d ≥ 2. We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair (Λ, Λ) of Λ and its derivative is Markovian) if and only if d = 2. In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.
We study the spectrum of the kinetic Brownian motion in the space of d × d Hermitian matrices, d ≥ 2. We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair (Λ, Λ) of Λ and its derivative is Markovian) if and only if d = 2. In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.
We consider the kernel of a hypoelliptic diffusion beyond the case of sub-ellipticity or polynomial coefficients. We get a full asymptotic expansion for small times, based on a Duhameltype comparison with an approximate polynomial kernel. As in the sub-elliptic case, some change of scale based on the geometry of some Lie brackets yields a non-trivial limit for the kernel as time goes to zero. Remarkably, a different scale is needed to observe a non-trivial large deviation principle. 24 7. Small time expansions 29 8. The macroscopic and mesoscopic scales 31 Appendix A. Qualitative regularity for the diffusion 36 Appendix B. Closed form for the Fourier transform 38 References 41
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