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 ${\mathcal D} \subset {\mathbb R}^d$ 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 ${\mathcal D}$, and we study the angle of intersection of the process with a $(d-1)$-dimensional manifold contained in ${\mathcal D}$.Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains

Normal aging and decision making: The role of motivation

The main argument of this review is that motivational development associated with normal aging affects decision making. With increasing age, the ratio of gains to losses becomes more and more unfavorable. Reflecting the increasing losses in resources, goal orientation changes from a predominant orientation towards gains in young adulthood to an increasingly stronger orientation towards the prevention of loss in older adulthood. As goals serve as reference points for the evaluation of decision outcomes, this change in goal orientation across adulthood might also affect decision making. The decision-making literature has recognized that choices are influenced by goal orientation. However, little research has been conducted on how goals influence the decision-making process in general and with regard to aging in particular. To date, findings on decision making and aging remain inconsistent and are in need of a developmental framework. With regard to applications, a better understanding of the aging decision maker can provide insight into how to improve communication efforts about issues like advance care planning, medical treatment, and housing options