On the time evolution of Bernstein processes associated with a class of parabolic equations

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved

On the Magnetic Prandtl Number Behavior of Accretion Disks

We investigate the behavior of the magnetic Prandtl number (ratio of microscopic viscosity to resistivity) for accretion sources. Generally this number is very small in standard accretion disk models, but can become larger than unity within $\sim 50$ Schwarzschild radii of the central mass. Recent numerical investigations suggest a marked dependence of the level of MHD turbulence on the value of the Prandtl number. Hence, black hole and neutron star accretors, i.e. compact X-ray sources, are affected. The astrophysical consequences of this could be significant, including a possible route to understanding the mysterious state changes that have long characterized these sources.Comment: 15 pages, 6 figures. Accepted for publication in the Astrophysical Journal (February 10, 2008 issue.) Minor changes from original submissio

Dewetting of solid films with substrate mediated evaporation

The dewetting dynamics of an ultrathin film is studied in the presence of evaporation - or reaction - of adatoms on the substrate. KMC simulations are in good agreement with an analytical model with diffusion, rim facetting, and substrate sublimation. As sublimation is increased, we find a transition from the usual dewetting regime where the front slows down with time, to a sublimation-controlled regime where the front velocity is approximately constant. The rim width exhibits an unexpected non-monotonous behavior, with a maximum in time.Comment: 6 pages, 6 figure