Viscosity methods giving uniqueness for martingale problems

Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $D$ to be nonsmooth and the direction of reflection to be degenerate. Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix

Models of granular ratchets

We study a general model of granular Brownian ratchet consisting of an asymmetric object moving on a line and surrounded by a two-dimensional granular gas, which in turn is coupled to an external random driving force. We discuss the two resulting Boltzmann equations describing the gas and the object in the dilute limit and obtain a closed system for the first few moments of the system velocity distributions. Predictions for the net ratchet drift, the variance of its velocity fluctuations and the transition rates in the Markovian limit, are compared to numerical simulations and a fair agreement is observed.Comment: 15 pages, 4 figures, to be published on Journal of Statistical Mechanics: Theory and Experiment

Jets and outflows in Radio Galaxies: implications for AGN feedback

One of the main debated astrophysical problems is the role of the AGN feedback in galaxy formation. It is known that massive black holes have a profound effect on the formation and evolution of galaxies, but how black holes and galaxies communicate is still an unsolved problem. For Radio Galaxies, feedback studies have mainly focused on jet/cavity systems in the most massive and X-ray luminous galaxy clusters. The recent high-resolution detection of warm absorbers in some Broad Line Radio Galaxies allow us to investigate the interplay between the nuclear engine and the surrounding medium from a different perspective. We report on the detection of warm absorbers in two Broad Line Radio Galaxies, 3C 382 and 3C 390.3, and discuss the physical and energetic properties of the absorbing gas. Finally, we attempt a comparison between radio-loud and radio-quiet outflows.Comment: To be published in the proceedings of High Energy Phenomena in Relativistic Outflows III (HEPRO III, IJMPCS). 4 pages, 2 figure

Protein accumulation in the endoplasmic reticulum as a non-equilibrium phase transition

Several neurological disorders are associated with the aggregation of aberrant proteins, often localized in intracellular organelles such as the endoplasmic reticulum. Here we study protein aggregation kinetics by mean-field reactions and three dimensional Monte carlo simulations of diffusion-limited aggregation of linear polymers in a confined space, representing the endoplasmic reticulum. By tuning the rates of protein production and degradation, we show that the system undergoes a non-equilibrium phase transition from a physiological phase with little or no polymer accumulation to a pathological phase characterized by persistent polymerization. A combination of external factors accumulating during the lifetime of a patient can thus slightly modify the phase transition control parameters, tipping the balance from a long symptomless lag phase to an accelerated pathological development. The model can be successfully used to interpret experimental data on amyloid-\b{eta} clearance from the central nervous system