Distributions of a particle’s position and their asymptotics in the q-deformed totally asymmetric zero range process with site dependent jumping rates

In this paper we study the probability distribution of the position of a tagged particle in the $q$-deformed Totally Asymmetric Zero Range Process ($q$-TAZRP) with site dependent jumping rates. For a finite particle system, it is derived from the transition probability previously obtained by Wang and Waugh. We also provide the probability distribution formula for a tagged particle in the $q$-TAZRP with the so-called step initial condition in which infinitely many particles occupy one single site and all other sites are unoccupied. For the $q$-TAZRP with step initial condition, we provide a Fredholm determinant representation for the probability distribution function of the position of a tagged particle, and moreover we obtain the limiting distribution function as the time goes to infinity. Our asymptotic result for $q$-TAZRP with step initial condition is comparable to the limiting distribution function obtained by Tracy and Widom for the $k$th leftmost particle in the asymmetric simple exclusion process with step initial condition (Theorem 2 in Tracy and Widom (2009)).     

Limit velocity for a driven particle in a random medium with mass aggregation

We study a one-dimensional infinite system of particles driven by a constant positive force F which acts only on the leftmost particle which is regarded as the tracer particle (t.p.). All other particles are field neutral, do not interact among themselves, and independently of each other with probability 0<p≤1 are either perfectly inelastic and “stick” to the t.p. after the first collision, or with probability 1−p are perfectly elastic, mechanically identical and have the same mass m. At initial time all particles are at rest, and the initial measure is such that the interparticle distances ξ_i's are i.i.d. r.v.'s. with absolutely continuous density. We show that for any value of the field F>0, the velocity of the t.p. converges to a limit value, which we compute.     

Continuity for the asymptotic shape in the frog model with random initial configurations

We consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. An interest of this model is how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time. Thus, in this paper, we prove continuity for the asymptotic shape in the law of the initial configuration.     

Fringe analysis of plane trees related to cutting and pruning

Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.     

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

A new approach to the cascade theory

The cascade theory of cosmic rays is investigated from a new point of view; we deal with the number of particles produced in a certain thickness of matter, the energy of each particle being greater than E at the point of its production. This new approach leads to an elegant asymptotic theory for large thicknesses and it is better adapted to the interpretation of experiments in nuclear emulsions.     

Examples of asymptotic completeness in translation invariant systems with an unbounded number of particles

A tagged particle interacting with a finite but unbounded number of particles is considered. Some examples of asymptotic completeness are given which are uniform in the number of particles.     

Asymptotic behavior of the wave function of three particles in a continuum

We study the wave function of a system of three particles in a continuum. The Faddeev equations are used to explicitly identify the singularities of the wave function in the momentum space. We obtain the asymptotic behavior of the wave function in the configuration space by calculating the asymptotic behavior of the Fourier transform of the wave function in the momentum space. Our attention is focused on configurations in which two particles are at a relatively small distance from each other while the third particle is significantly remote from the center of mass of the pair. We show that the coordinate asymptotic form of the wave function for such a configuration contains scattered waves of a new type in addition to the standard terms. We use the obtained exact data concerning the coordinate asymptotic form of the wave function to critically analyze the multiplicative ansatz used in several works to describe systems of three particles in a continuum.     

Spatially homogeneous solutions of the Vlasov–Nordström–Fokker–Planck system

The Vlasov–Nordström–Fokker–Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the spatially-homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions. Additionally, we study the long time asymptotic behavior of the system and prove that even in the absence of friction, solutions possess a non-trivial asymptotic profile. An exact formula for the long time limit of the particle density is derived in the ultra-relativistic case.     

Immortal particle for a catalytic branching process

We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth coefficients for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive are given. In the Brownian motions case the domain of evolution can be non-smooth, including Lipschitz, with integrable Martin kernel. The results are valid for an arbitrary number of particles and non-uniform redistribution after branching. Additionally, with probability one, it is shown that only one ancestry line survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times.     

Particle Approximations of the Score and Observed Information Matrix for Parameter Estimation in State–Space Models With Linear Computational Cost

Poyiadjis, Doucet, and Singh showed how particle methods can be used to estimate both the score and the observed information matrix for state–space models. These methods either suffer from a computational cost that is quadratic in the number of particles, or produce estimates whose variance increases quadratically with the amount of data. This article introduces an alternative approach for estimating these terms at a computational cost that is linear in the number of particles. The method is derived using a combination of kernel density estimation, to avoid the particle degeneracy that causes the quadratically increasing variance, and Rao–Blackwellization. Crucially, we show the method is robust to the choice of bandwidth within the kernel density estimation, as it has good asymptotic properties regardless of this choice. Our estimates of the score and observed information matrix can be used within both online and batch procedures for estimating parameters for state–space models. Empirical results show improved parameter estimates compared to existing methods at a significantly reduced computational cost. Supplementary materials including code are available.     

Surviving particles for subcritical branching processes in random environment

The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives a.s$a.s$. These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.     

Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.     

球面介质作用下的测度值分枝过程

我们考虑空间上一粒子系统，当其受到分布于求面上的介质作进行粒子分枝和衍生，产生新了体，而新粒子仍按原粒子的运动规则继续空间运动。通过合理的假设和极限过程，粒子在空间的散布一测度值分枝过程来刻划。     

Forward and backward stochastic differential equations with normal constraints in law

In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding “normal” vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.     

Branching Brownian Motion with Catalytic Branching at the Origin

We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate β>0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above λt at time t, for λ>0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.     

Effect of spherically symmetric mass flow from the surface of a particle on the force of interaction with a plane surface

A stationary velocity field of the flow of a gaseous medium generated by uniform radial injection from the surface of a spherical particle near a wall is considered in the Stokes' approximation. Bispherical coordinates are used to write the expression for the stream function. A formula is obtained for the force acting on the spherical particle when there is an arbitrary mass flow from its surface, generalizing earlier results /1, 2/. An expression for the force acting on the particle is obtained for the case of spherically symmetric injection from the surface of the particle, and asymptotic formulas at short and long distances from the wall are studied.

An analogous problem concerning the forces of interaction between two spherical particles of the same radius, when uniform injection of equal intensity takes place from their surfaces, is discussed. This is equivalent to the problem of the interaction of a spherical particle with a free surface. A general expression for the force of interaction, and its asymptotic forms for short and long distances, are obtained.     

Chebyshev Wavelet Procedure for Solving FLDEs

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. This setting covers, for instance, pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge in characterising the boundary behaviour is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via \(\Gamma \)-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.     

The Infrared Problem in QED: A Lesson from a Model with Coulomb Interaction and Realistic Photon Emission

The scattering of photons and heavy classical Coulomb interacting particles, with realistic particle–photon interaction (without particle recoil) is studied adopting the Koopman formulation for the particles. The model is translation invariant and allows for a complete control of the Dollard strategy devised by Kulish–Faddeev and Rohrlich (KFR) for QED: in the adiabatic formulation, the Møller operators exist as strong limits and interpolate between the dynamics and a non-free asymptotic dynamics, which is a unitary group; the S-matrix is non-trivial and exhibits the factorization of all the infrared divergences. The implications of the KFR strategy on the open questions of the LSZ asymptotic limits in QED are derived in the field theory version of the model, with the charged particles described by second quantized fields: i) asymptotic limits of the charged fields, \({\Psi_{{\rm out}/{\rm in}}(x)}\), are obtained as strong limits of modified LSZ formulas, with corrections given by a Coulomb phase operator and an exponential of the photon field; ii) free asymptotic electromagnetic fields, \({B_{{\rm out}/{\rm in}}(x)}\), are given by the massless LSZ formula, as in Buchholz approach;   iii) the asymptotic field algebras are a semidirect product of the canonical algebras generated by \({B_{{\rm out}/{\rm in}}}\), \({\Psi_{{\rm out}/{\rm in}}}\);   iv) on the asymptotic spaces, the Hamiltonian is the sum of the free (commuting) Hamiltonians of \({B_{{\rm out}/{\rm in}}}\), \({\Psi_{{\rm out}/{\rm in}}}\) and the same holds for the generators of the space translations.     

Strong and weak convergence of the population size in a supercritical catalytic branching process

A general model of a catalytic branching process (CBP) with any number of catalysis centers in a discrete space is studied. The asymptotic (in time) behavior of the total number of particles and of the local particle numbers is investigated. The problems of finding the global and local extinction probabilities are solved. Necessary and sufficient conditions are established for the phase of pure global survival and strong local survival. Under wide conditions, limit theorems for the normalized total and local particle numbers in supercritical CBP are proved in the sense of almost sure convergence, as well as with respect to convergence in distribution. Generalizations of a number of previous results are obtained as well.