REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

Operation Everest II and the 1978 Habeler/Messner ascent of Everest without bottled O2: what might they have in common?

In 1978, Peter Habeler and Reinhold Messner climbed Everest without supplemental O2. Subsequently, Oelz et al. (Oelz O, Howald H, Di Prampero PE, Hoppeler H, Claassen H, Jenni R, Bühlmann A, Ferretti G, Brückner JC, Veicsteinas A, Gussoni M, Cerretelli P. J Appl Physiol (1985) 60: 1734–1742, 1986) assessed their cardiopulmonary function, finding no advantageous physiological attributes to explain their success, and leading West (West JB. High Life: A History of High-Altitude Physiology and Medicine. New York: Oxford University, 1998) to suggest that grit and determination were more important. In 1985, Charlie Houston, John Sutton, and Al Cymerman hosted a scientific project assessing a simulated ascent of Everest (OE II) at the U.S. Army Research Institute of Environmental Medicine. Included were measurements of O2 transport. In particular, mixed venous Po2 was measured at/near maximal exercise, for calculating pulmonary O2-diffusing capacity. A serendipitous observation was made: while both V̇o2max and mixed venous Po2 fell with altitude (as expected), it was how they fell—in direct proportion—that was remarkable. It later became clear that this reflected diffusion limitation of O2 transport from muscle microvessels to the mitochondria, and that this last step in O2 transport plays a major role in limiting V̇o2max. Thus, how Habeler and Messner made it up Everest without bottled O2 and no special cardiopulmonary attributes might be explained if their muscle O2-diffusing capacity, which depends largely on muscle capillarity, was unusually high. Oelz et al. mention that muscle capillary density was substantially—40%—above normal, but did not suggest that this accounted for the climbersʼ success. Therefore, high muscle capillarity, enhancing diffusive unloading of O2, may have been a major enabling physiological attribute for Habeler and Messner and that OE II, by chance, played a key role in bringing this to light.

Invariant density estimation for a reflected diffusion using an Euler scheme

We give an explicit error bound between the invariant density of an elliptic reflected diffusion in a smooth compact domain and the kernel estimator built on the symmetric Euler scheme introduced in [

Reducing Response Time in Fork-Join Systems under Heavy Traffic Via Imbalance Control

We consider the problem of reducing the response time of fork-join systems by maintaining the workload balanced among the processing stations. The general problem of modeling and finding an optimal policy that reduces imbalance is quite difficult. In order to circumvent this difficulty, the heavy traffic approach is taken, and the system dynamics are approximated by a reflected diffusion process. This way, the problem of finding an optimal balancing policy that reduces workload imbalance is set as a stochastic optimal control problem, for which numerical methods are available. Some numerical experiments are presented, where the control problem is solved numerically and applied to a simulation. The results indicate that the response time of the controlled system is reduced significantly using the devised control.

Reducing Response Time in Fork-Join Systems under Heavy Traffic Via Imbalance Control

We consider the problem of reducing the response time of fork-join systems by maintaining the workload balanced among the processing stations. The general problem of modeling and finding an optimal policy that reduces imbalance is quite difficult. In order to circumvent this difficulty, the heavy traffic approach is taken, and the system dynamics are approximated by a reflected diffusion process. This way, the problem of finding an optimal balancing policy that reduces workload imbalance is set as a stochastic optimal control problem, for which numerical methods are available. Some numerical experiments are presented, where the control problem is solved numerically and applied to a simulation. The results indicate that the response time of the controlled system is reduced significantly using the devised control.