5 research outputs found
Billiards in a general domain with random reflections
We study stochastic billiards on general tables: a particle moves according
to its constant velocity inside some domain until it hits the boundary and bounces randomly inside according to some
reflection law. We assume that the boundary of the domain is locally Lipschitz
and almost everywhere continuously differentiable. The angle of the outgoing
velocity with the inner normal vector has a specified, absolutely continuous
density. We construct the discrete time and the continuous time processes
recording the sequence of hitting points on the boundary and the pair
location/velocity. We mainly focus on the case of bounded domains. Then, we
prove exponential ergodicity of these two Markov processes, we study their
invariant distribution and their normal (Gaussian) fluctuations. Of particular
interest is the case of the cosine reflection law: the stationary distributions
for the two processes are uniform in this case, the discrete time chain is
reversible though the continuous time process is quasi-reversible. Also in this
case, we give a natural construction of a chord "picked at random" in
, and we study the angle of intersection of the process with a
-dimensional manifold contained in .Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics
and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains
Estimates for the Syracuse problem via a probabilistic model
In the paper we employ a simple stochastic model for the 'Syracuse problem' (aka '3x+1 problem') to get estimates for the 'average behaviour' of the trajectories of the original deterministic dynamical system. The use of the model is supported not only by certain similarities between the governing rules in the systems, but also by a qualitative estimate of the rate of approximation (Theorem 2). From the model, we derive explicit formulae for the asymptotic densities of some sets of interest for the original sequence. We also approximate the asymptotic distributions for the 'stopping times' (times till absorption in the only known cycle) of the original system and give numerical illustrations to our results. (orig.)Available from TIB Hannover: RR 9140(99-01) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman