Properties of solutions of stochastic differential equations driven by the G-Brownian motion

In this paper, we study the differentiability of solutions of stochastic differential equations driven by the $G$-Brownian motion with respect to the initial data and the parameter. In addition, the stability of solutions of stochastic differential equations driven by the $G$-Brownian motion is obtained

Chaotic cold accretion onto black holes

Using 3D AMR simulations, linking the 50 kpc to the sub-pc scales over the course of 40 Myr, we systematically relax the classic Bondi assumptions in a typical galaxy hosting a SMBH. In the realistic scenario, where the hot gas is cooling, while heated and stirred on large scales, the accretion rate is boosted up to two orders of magnitude compared with the Bondi prediction. The cause is the nonlinear growth of thermal instabilities, leading to the condensation of cold clouds and filaments when t_cool/t_ff < 10. Subsonic turbulence of just over 100 km/s (M > 0.2) induces the formation of thermal instabilities, even in the absence of heating, while in the transonic regime turbulent dissipation inhibits their growth (t_turb/t_cool < 1). When heating restores global thermodynamic balance, the formation of the multiphase medium is violent, and the mode of accretion is fully cold and chaotic. The recurrent collisions and tidal forces between clouds, filaments and the central clumpy torus promote angular momentum cancellation, hence boosting accretion. On sub-pc scales the clouds are channelled to the very centre via a funnel. A good approximation to the accretion rate is the cooling rate, which can be used as subgrid model, physically reproducing the boost factor of 100 required by cosmological simulations, while accounting for fluctuations. Chaotic cold accretion may be common in many systems, such as hot galactic halos, groups, and clusters, generating high-velocity clouds and strong variations of the AGN luminosity and jet orientation. In this mode, the black hole can quickly react to the state of the entire host galaxy, leading to efficient self-regulated AGN feedback and the symbiotic Magorrian relation. During phases of overheating, the hot mode becomes the single channel of accretion (with a different cuspy temperature profile), though strongly suppressed by turbulence.Comment: Accepted by MNRAS: added comments and references. Your feedback is welcom

Some properties on GG-evaluation and its applications to GG-martingale decomposition

In this article, a sublinear expectation induced by $G$-expectation is introduced, which is called $G$-evaluation for convenience. As an application, we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the decomposition theorem holds and any $\beta>1$ integrable symmetric $G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t $G$-Brownian motion. As a byproduct, we prove a regular property for $G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous versionComment: 22 page

AGN Feedback and Bimodality in Cluster Core Entropy

We investigate a series of steady-state models of galaxy clusters, in which the hot intracluster gas is efficiently heated by active galactic nucleus (AGN) feedback and thermal conduction, and in which the mass accretion rates are highly reduced compared to those predicted by the standard cooling flow models. We perform a global Lagrangian stability analysis. We show for the first time that the global radial instability in cool core clusters can be suppressed by the AGN feedback mechanism, provided that the feedback efficiency exceeds a critical lower limit. Furthermore, our analysis naturally shows that the clusters can exist in two distinct forms. Globally stable clusters are expected to have either: 1) cool cores stabilized by both AGN feedback and conduction, or 2) non-cool cores stabilized primarily by conduction. Intermediate central temperatures typically lead to globally unstable solutions. This bimodality is consistent with the recently observed anticorrelation between the flatness of the temperature profiles and the AGN activity (Dunn & Fabian 2008) and the observation by Rafferty et al. (2008) that the shorter central cooling times tend to correspond to significantly younger AGN X-ray cavities.Comment: 4 pages, to appear in the proceedings of "The Monster's Fiery Breath: Feedback in Galaxies, Groups, and Clusters", Eds. Sebastian Heinz, Eric Wilcots (AIP conference series

On the threshold-width of graphs

The GG-width of a class of graphs GG is defined as follows. A graph G has GG-width k if there are k independent sets N1,...,Nk in G such that G can be embedded into a graph H in GG such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k are characterized by a finite collection of forbidden induced subgraphs

Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators

The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov equation and to stochastic optimal control

An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation

The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation