International audienceWe consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle T 1 M of a Riemannian manifold (M, g), collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter σ quantifying the size of the noise. Projection on M of these processes provides random C 1 paths in M. We show, both qualitively and quantitatively, that the laws of these M-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter σ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when σ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms

Two similar Minkowskian diffusions have been considered, on one hand by Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR], [DR]), and on the other hand by Dunkel and Hänggi ([DH1], [DH2]). We address here two questions, asked in [DR] and in ([DH1], [DH2]) respectively, about the asymptotic behaviour of such diffusions. More generally, we establish a central limit theorem for a class of Minkowskian diffusions, to which the two above ones belong. As a consequence, we correct a partially wrong guess in [DH1].

We consider random trigonometric polynomials of the form f n (x, y) = 1≤k,l≤n a k,l cos(kx) cos(ly), where the entries (a k,l ) k,l≥1 are i.i.d. random variables that are centered with unit variance. We investigate the length K (f n ) of the nodal set Z K (f n ) of the zeros of f n that belong to a compact set K ⊂ R 2 . We first establish a local universality result, namely we prove that, as n goes to infinity, the sequence of random variables n K/n (f n ) converges in distribution to a universal limit which does not depend on the particular law of the entries. We then show that at a macroscopic scale, the expectation of [0,π] 2 (f n )/n also converges to an universal limit. Our approach provides two main byproducts: (i) a general result regarding the continuity of the volume of the nodal sets with respect to C 1 -convergence which refines previous findings of [RS01, IK16, ADL+ 15] and (ii) a new strategy for proving small ball estimates in random trigonometric models, providing in turn uniform local controls of the nodal volumes.