Symmetric Jump Processes and their Heat Kernel Estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Holder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.Comment: To appear in Science in China Series A: Mathematic

The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr\"{o}dinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr\"{o}dinger evolution is analyzed in detail. The relativistic covariance of related waveComment: Latex fil

Structure of shocks in Burgers turbulence with L\'evy noise initial data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by L\'evy noise, or equivalently when the initial potential is a two-sided L\'evy process $\psi_0$. When $\psi_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $\psi_0$ is abrupt we show that the shock structure is discrete. When $\psi_0$ is eroded we show that there are no rarefaction intervals.Comment: 22 page

Newly uncovered physics of MHD instabilities using 2-D electron cyclotron emission imaging system in toroidal plasmas

Validation of physics models using the newly uncovered physics with a 2-D electron cyclotron emission imaging (ECEi) system for magnetic fusion plasmas has either enhanced the confidence or substantially improved the modeling capability. The discarded "full reconnection model" in sawtooth instability is vindicated and established that symmetry and magnetic shear of the 1/1 kink mode are critical parameters in sawtooth instability. For the 2/1 instability, it is demonstrated that the 2-D data can determine critical physics parameters with a high confidence and the measured anisotropic distribution of the turbulence and its flow in presence of the 2/1 island is validated by the modelled potential and gyro-kinetic calculation. The validation process of the measured reversed-shear Alfveneigenmode (RSAE) structures has improved deficiencies of prior models. The 2-D images of internal structure of the ELMs and turbulence induced by the resonant magnetic perturbation (RMP) have provided an opportunity to establish firm physics basis of the ELM instability and role of RMPs. The importance of symmetry in determining the reconnection time scale and role of magnetic shear of the 1/1 kink mode in sawtooth instability may be relevant to the underlying physics of the violent kink instability of the filament ropes in a solar flare

The In Vivo Kinetics of RNA Polymerase II Elongation during Co-Transcriptional Splicing

Kinetic analysis shows that RNA polymerase elongation kinetics are not modulated by co-transcriptional splicing and that post-transcriptional splicing can proceed at the site of transcription without the presence of the polymerase

Nuclear Organization and Dynamics of 7SK RNA in Regulating Gene Expression

We have identified 7SK RNA to be enriched in nuclear speckles. Knock-down of 7SK results in the mislocalization of nuclear speckle constituents, and the transcriptional up-regulation of a reporter gene locus. 7SK RNA transiently associates with the locus upon transcriptional down-regulation correlating with the displacement of pTEF-b

In Vivo Monitoring of mRNA Movement in Drosophila Body Wall Muscle Cells Reveals the Presence of Myofiber Domains

Background: In skeletal muscle each muscle cell, commonly called myofiber, is actually a large syncytium containing numerous nuclei. Experiments in fixed myofibers show that mRNAs remain localized around the nuclei in which they are produced. Methodology/Principal Findings: In this study we generated transgenic flies that allowed us to investigate the movement of mRNAs in body wall myofibers of living Drosophila embryos. We determined the dynamic properties of GFP-tagged mRNAs using in vivo confocal imaging and photobleaching techniques and found that the GFP-tagged mRNAs are not free to move throughout myofibers. The restricted movement indicated that body wall myofibers consist of three domains. The exchange of mRNAs between the domains is relatively slow, but the GFP-tagged mRNAs move rapidly within these domains. One domain is located at the centre of the cell and is surrounded by nuclei while the other two domains are located at either end of the fiber. To move between these domains mRNAs have to travel past centrally located nuclei. Conclusions/Significance: These data suggest that the domains made visible in our experiments result from prolonged interactions with as yet undefined structures close to the nuclei that prevent GFP-tagged mRNAs from rapidly moving between the domains. This could be of significant importance for the treatment of myopathies using regenerative cellbase