Homogenisation for anisotropic kinetic random motions

Abstract

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.

Kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold

“…This statement generalizes Proposition 1.1 of [Per18] to the present infinite dimensional setting. The above description of the invariant measure µ as an image measure under the projection map actually coincides with the finite dimensional description given in the latter reference.…”

“…The statement of Proposition 2.5 follows then from the conclusion of Dedecker and Merlevède convergence result. The identification of the covariance (7) is a consequence of the corresponding statement, Proposition 3.4, in the finite dimensional setting of [Per18].…”

“…As in the finite dimensional anisotropic case treated in [Per18], it is actually easier to work with an H-valued lift of this S-valued process. We introduce for that purpose the process (u σ t ) t≥0 solution of the Stratonovich stochastic differential equation…”

“…In [ABT15], Angst, Bailleul and Tardif gave the most general result, assuming only geodesic and stochastic completeness, using rough paths theory as a working horse to transport a rough path convergence result about kinetic Brownian motion in R d to the manifold setting. See also [Li18] for further results in homogeneous spaces, and [Per18] for a generalization of the homogenization result of [ABT15] to anisotropic kinetic Brownian motion, or more general Markov processes on T 1 M . Note that the dynamically obvious convergence of the unrescaled kinetic Brownian motion to the geodesic motion has been studied from the spectral point of view in [Dro17], for compact manifolds with negative curvature, showing that the L 2 spectrum of the generator of the unrescaled kinetic Brownian motion converges to the Pollicott-Ruelle resonances of M .…”

Kinetic Dyson Brownian motion

“…A self-contained proof by the same author appeared later in [Li16]. This result was generalised by Angst, Bailleul and Tardif [ABT15] and the author [Per20]. Kinetic Brownian motion has been given different names in the literature, for instance velocity spherical Brownian motion by Baudoin and Tardif [BT18] or circular Langevin diffusion by Franchi [Fra15] in the context of heat kernels.…”