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Our starting point is a formal linear stochastic differential equation of first order (higher order equations can be transformed to systems of these)
where I, a, W are stochastic functions with and analogously for a and W. I, a, and W are allowed to depend on the element ω of a set Ω in which a probability measure is defined in the usual way (see e.g. Doob, 1953; de Witt-Morette, 1981). To get a solution of eq.(1) for the mean intensity we treat the problem according the Reynolds averaging technique in the usual manner : The stochastic equation is changed into an infinte hierarchical system of equations for the correlations.
We present a first exploratory investigation of the dynamical evolution of a dusty stellar wind envelope along the upper AGB and its transformation into a planetary nebula. We find the existence of AGB stars with detached shells to be a natural consequence of the mass-loss variations during a thermal pulse. It is also demonstrated that due to the large dynamical effects caused by the ionizing radiation field and the fast wind of the central star, it is impossible to deduce the AGB mass loss history from the planetary's density and velocity distribution. The structure of the halo, however, is still determined by the AGB mass loss history. The rapid decline of mass loss expected in the aftermath of thermal pulses leads to extended shells of low densities and explains halos with sharp boundaries.
Based on the mass-loss description developed by Blöcker (1995, A&A, 297, 727), we present first exploratory computations of the dynamical evolution of a dusty stellar wind envelope around an intermediate mass star during the last 300 000 years on the AGB and its transformation into a planetary nebula during the following 5 000 years of post-AGB evolution. To model the dynamics of the cool dusty envelope, we used a two-component (gas/dust) 1D radiation hydrodynamics code which computes the radiation pressure on dust grains and the structure of the envelope in a self-consistent way, including the variable frictional coupling between dust and gas. The grains are either carbon or oxygen based and of single size and spherical shape.
Using a spherical aberration (Cs)-corrected scanning transmission electron microscopy (STEM) and electron energy-loss spectroscopy (EELS), we investigated a 6° low-angle  tilt grain boundary in SrTiO3. The enhanced spatial resolution of the aberration corrector leads to the observation of a number of structural variations in the edge dislocations along the grain boundary that neither resemble the standard edge dislocations nor partial dislocations for SrTiO3. Although there appear to be many variants in the structure that can be interpreted as compositional effects, three main classes of core structure are found to be prominent. From EELS analysis, these classifications seem to be related to Sr deficiencies, with the final variety of the cores being consistent with an embedded TiOx rocksalt-like structure.