Jürgen Angst

Central limit theorem for a class of relativistic diffusions

Two similar Minkowskian diffusions have been considered, on one hand by Debbasch and co-workers [J. Math. Phys. 40, 2891 (2001); Eur. Phys. J. 19, 37 (2001); 23, 487 (2001); J. Stat. Phys. 88, 945 (1997); 90, 1179 (1998)], and on the other hand by Dunkel and Hanggi [Phys. Rev. E 71, 016124 (2005); 72, 036106 (2005)]. We address here two questions, asked by Debbasch and Rivet [J. Stat. Phys. 90, 1179 (1998)] and by Dunkel and Hanggi [Phys. Rev. E 71, 016124 (2005); 72, 036106 (2005)], respectively, about the asymptotic behavior 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 by Dunkel and Hanggi [Phys. Rev. E 71, 016124 (2005)].

Trends to equilibrium for a class of relativistic diffusions

A large class C of relativistic diffusions with values in the phase-space of special relativity was introduced by Angst and Franchi [J. Math. Phys. 48(8), 083101 (2007)] in order to answer some open questions concerning the asymptotic behavior of two examples of such processes. In particular, the equilibrium measures of these diffusions were explicitly computed, and their hydrodynamic limit was shown to be Brownian. In this paper, we address the question of the trends to equilibrium of the momentum component of the diffusions of the whole class C. We show the existence of a spectral gap using the Lyapounov method and deduce the exponential decay of the distance to equilibrium in L2−norm and in total variation. A similar result was obtained recently by Calogero [e-print arXiv:1009.5086v2] for a particular process of the class C.

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

We present a method that allows, under suitable equivariance and regularity conditions, to determine the Poisson boundary of a diffusion starting from the Poisson boundary of a sub-diffusion of the original one. We then give two examples of application of this devissage method. Namely, we first recover the classical result that the Poisson boundary of Brownian motion on a rotationally symmetric manifolds is generated by its escape angle, and we then give an “elementary” probabilistic proof of the delicate result of Bailleul (Probab Theory Relat Fields 141(1–2):283-329, 2008), i.e. the determination of the Poisson boundary of the relativistic Brownian motion in Minkowski space-time.

A weak Cramér condition and application to Edgeworth expansions

We introduce a new, weak Cramer condition on the characteristic function of a random vector which does not only hold for all continuous distributions but also for discrete (non-lattice) ones in a generic sense. We then prove that the normalized sum of independent random vectors satisfying this new condition automatically verifies some small ball estimates and admits a valid Edgeworth expansion for the Kolmogorov metric. The latter results therefore extend the well known theory of Edgeworth expansion under the standard Cramer condition, to distributions that are purely discrete.