Statistical Analysis of the First Passage Path Ensemble of Jump Processes

The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas.     

Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the solution of the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative entropy density per unit time.     

Pricing on electricity market based on coupled-continuous-time-random-walk concept

In this paper we propose a model of electricity market based on the forward rate dynamics described by a diffusion with jumps as a generalization of the classical diffusion approach. We consider jump components resulting from a coupled continuous-time random walk (CTRW) with jump lengths proportional to the corresponding inter-jump time intervals. In the framework of the model we derive a formula for the EURO-price of a standard European call option, showing applicability of CTRW processes for pricing of financial instruments. The result, obtained by an advance theory of semimartingales, is an essential extension of the pricing formula derived in the classical diffusion model of the forward rate dynamics. It indicates an influence of both, the continuous and the jump parts of the forward rate process on the option price.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

Extremal behavior of a coupled continuous time random walk

Coupled continuous time random walks (CTRWs) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density governing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.     

Transition events fluctuations in jump markovian dynamics. A general approach to Šolc's problem

We introduce a new formalism for computing the moments of transition events for nonhomogeneous Markov jump processes. Our method is applied directly to the master equation and does not involve the use of diffusion approximation. The general theory is applied to produce exact expressions for means and dispersions. For time homogeneous Markov processes with a finite number of connected states we are able to prove that both means and dispersions asymptotically increase linearly in time.     

Facilitated asymmetric exclusion

We introduce a class of facilitated asymmetric exclusion processes in which particles are pushed by neighbors from behind. For the simplest version in which a particle can hop to its vacant right neighbor only if its left neighbor is occupied, we determine the steady-state current and the distribution of cluster sizes on a ring. We show that an initial density downstep develops into a rarefaction wave that can have a jump discontinuity at the leading edge, while an upstep results in a shock wave. This unexpected rarefaction wave discontinuity occurs generally for facilitated exclusion processes.     

Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model

Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the option prices themselves, so that jump processes are taken into account in conjunction with the usual drift and diffusion terms. In particular, we focus on stochastic volatility models, such as the exponential Vasicek model, and extend the pricing formulas and propagator of this model to incorporate jump diffusion with a given jump size distribution. This model is of importance to include non-Gaussian fluctuations beyond the Black-Scholes model, and moreover yields a lognormal distribution of the volatilities, in agreement with results from superstatistical analysis. The results obtained in the present formalism are checked with Monte Carlo simulations.     

Jumps in icosahedral quasicrystals

Atomic jumps in icosahedral (AlCu)Li quasicrystals and related structures have been studied by molecular dynamics simulations. In quasicrystalline structures jumps exists with jump vectors much shorter than an average nearest neighbor distance. This is a consequence of the phasonic degree of freedom. The jumps therefore are called flips and the sites connected by the jump vector are denoted alternative positions. We find that the atoms in the quasicrystal structures studied here do not flip to alternative positions as proposed and observed in decagonal or dodecagonal quasicrystals but jump to sites which are at least an ordinary interatomic distance apart. Furthermore we observe two diffusion regimes: below about 55% of the melting temperature only small (AlCu) atoms carry out ring processes whereas at higher temperatures both kinds of atoms contribute to long-range diffusion. Received 21 July 1999     

Stochastic motion of a charged particle in a magnetic field: I Classical treatment

We study the dissipative, classical dynamics of a charged particle in the presence of a magnetic field. Two stochastic models are employed, and a comparative analysis is made, one based on diffusion processes and the other on jump processes. In the literature on collision-broadening of spectral lines, these processes go under the epithet of weak-collision model and Boltzmann-Lorentz model, respectively. We apply our model calculation to investigate the effect of magnetic field on the collision-broadened spectral lines, when the emitter carries an electrical charge. The spectral lines show narrowing as the magnetic field is increased, the narrowing being sharper in the Boltzmann-Lorentz model than in the weak collision model.     

Dynamic pressure jump in barrier-discharge excilamps

Data for the pressure jump are used to study thermodynamic processes in barrier-discharge excilamps under the conditions of a diffuse (uniform) discharge and microdischarges. It is shown that a linear relationship between the pressure jump and excilamp radiation intensity can help to control the radiation power.     

Mechanics of the turbulent-nonturbulent interface of a jet

We report the results of an experimental investigation of the mechanics and transport processes at the bounding interface between the turbulent and nonturbulent regions of flow in a turbulent jet, which shows the existence of a finite jump in the tangential velocity at the interface. This is associated with small-scale eddying motion at the outward propagating interface (nibbling) by which irrotational fluid becomes turbulent, and this implies that large-scale engulfment is not the dominant entrainment process. Interpretation of the jump as a singular structure yields an essential and significant contribution to the mean shear in the jet mixing region. Finally, our observations provide a justification for Prandtl's original hypothesis of a constant eddy viscosity in the nonturbulent outer jet region.     

Optimal disorder for segregation in annealed small worlds

We study a model for microscopic segregation in a homogeneous system of particles moving on a one-dimensional lattice. Particles tend to separate from each other, and evolution ceases when at least one empty site is found between any two particles. Motion is a mixture of diffusion to nearest-neighbour sites and long-range jumps, known as annealed small-world propagation. The long-range jump probability plays the role of the small-world disorder. We show that there is an optimal value of this probability, for which the segregation process is fastest. Moreover, above a critical probability, the time needed to reach a fully segregated state diverges for asymptotically large systems. These special values of the long-range jump probability depend crucially on the particle density. Our system is a novel example of the rare dynamical processes with critical behaviour at a finite value of the small-world disorder.     

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.     

Fractal and lacunary stochastic processes

Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently shortranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Lévy or stable processes. A one-dimensional walk with a selfsimilar jump distribution (the Weierstrass random walk) and its higherdimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the Hausdorff-Besicovitch dimension of stable processes. The Weierstrass random walk is lacunary (has gaps in the set of allowed steps) and its characteristic function is Weierstrass' non-differentiable function. Other lacunary random walks with characteristic functions related to Riemann's zeta function and certain numbertheoretic functions have very interesting analytic structure.     

含多层InAs量子点的双肖特基势垒二极管输运特性研究

研究了含多层InAs量子点结构的双肖特基势垒的电流输运特性,观察到了量子点的电子存储效应及其对电流的调制现象、电流多稳态现象和零点电压漂移现象.因为多量子点之间存在耦合作用,造成器件中的很多亚稳态.通过器件的输运特性显示出比含单层量子点器件更复杂的结果.随着外加电压的变化,器件经历很多弛豫过程.这些弛豫过程在电流电压曲线中造成很多电流跳跃结构和各种噪声结构 关键词： 多量子点 迟滞现象 单电子过程     

Inside Singularity Sets of Random Gibbs Measures

We evaluate the scale at which the multifractal structure of some random Gibbs measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar Gibbs measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes.