Interacting particle systems approximations of the Kushner-Stratonovitch equation

1999 ◽  
Vol 31 (3) ◽  
pp. 819-838 ◽  
Author(s):  
D. Crişan ◽  
P. Del Moral ◽  
T. J. Lyons
Keyword(s):  
Continuous Time ◽  
Order Of Convergence ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Interacting Particle System ◽  
Order Convergence ◽  
Interacting Particle ◽  
Convergence Results ◽  
Filtering Problem

In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order convergence results.

Interacting particle systems approximations of the Kushner-Stratonovitch equation

1999 ◽  
Vol 31 (03) ◽  
pp. 819-838 ◽  
Author(s):  
D. Crişan ◽  
P. Del Moral ◽  
T. J. Lyons
Keyword(s):  
Continuous Time ◽  
Order Of Convergence ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Interacting Particle System ◽  
Order Convergence ◽  
Interacting Particle ◽  
Convergence Results ◽  
Filtering Problem

In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order convergence results.

Estimation of critical values in interacting particle systems

2000 ◽  
Vol 37 (01) ◽  
pp. 118-125
Author(s):  
Raúl Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Computer Simulation ◽  
Asymptotic Behaviour ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Critical Values ◽  
Interacting Particle System ◽  
Absorption Time ◽  
Interacting Particle ◽  
Absorbing State

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.

SELF-PROPELLED INTERACTING PARTICLE SYSTEMS WITH ROOSTING FORCE

2010 ◽  
Vol 20 (supp01) ◽  
pp. 1533-1552 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
AXEL KLAR ◽  
STEPHAN MARTIN ◽  
SUDARSHAN TIWARI
Keyword(s):  
Stochastic Model ◽  
Collective Behavior ◽  
Food Source ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Mean Field ◽  
Particle Systems ◽  
Interacting Particle System ◽  
Hydrodynamic Equations ◽  
Interacting Particle

We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.

Estimation of critical values in interacting particle systems

2000 ◽  
Vol 37 (1) ◽  
pp. 118-125
Author(s):  
Raúl Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Computer Simulation ◽  
Asymptotic Behaviour ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Critical Values ◽  
Interacting Particle System ◽  
Absorption Time ◽  
Interacting Particle ◽  
Absorbing State

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.

A note on the extinction rates of some birth–death processes

1979 ◽  
Vol 16 (04) ◽  
pp. 897-902
Author(s):  
Maury Bramson ◽  
David Griffeath
Keyword(s):  
Continuous Time ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Holding Time ◽  
Constant Multiple ◽  
Extinction Rates ◽  
Interacting Particle ◽  
Positive Integers ◽  
Birth Death

In this note we are concerned with the rate of extinction of certain continuous-time birth-death processes on the positive integers with absorption at 0. The class we deal with includes birth-death processes with mean holding time h(i) at i such that h (i)∼ i–α as i →∞, 0 ≦ α< 1. In general, our result estimates to within a constant multiple the probability of non-extinction by time t. For h(i)∼ i–α , the result states that the probability of non-extinction is of order t−1/(2-α) We give an application to interacting particle systems.

Explicit sufficient invariants for an interacting particle system

1998 ◽  
Vol 35 (3) ◽  
pp. 633-641 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Colin Mallows ◽  
Larry Shepp
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
The Other ◽  
New Class ◽  
Interacting Particle ◽  
Death State ◽  
Number Of Particles

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are Ni(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = Ni(∞), as a function of Ni(0).We are able to obtain, for some special graphs, the limiting distribution of Ni if the total number of particles N → ∞ in such a way that the fraction, Ni(0)/S = ξi, at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S2, the two-leaf star which has three vertices and two edges.

Explicit sufficient invariants for an interacting particle system

1998 ◽  
Vol 35 (03) ◽  
pp. 633-641 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Colin Mallows ◽  
Larry Shepp
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
The Other ◽  
New Class ◽  
Interacting Particle ◽  
Death State ◽  
Number Of Particles

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η i = N i (∞), as a function of N i (0). We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξ i , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.

EXTENDED APPLICATION OF CONSTRUCTIVE CRITERIA FOR ERGODICITY OF INTERACTING PARTICLE SYSTEMS

1996 ◽  
Vol 07 (01) ◽  
pp. 1-18 ◽  
Author(s):  
HANS DE JONG ◽  
CHRISTIAN MAES
Keyword(s):  
Cellular Automata ◽  
Discrete Time ◽  
Continuous Time ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Probabilistic Cellular Automata ◽  
Interacting Particle ◽  
Spin Flip ◽  
Extended Application ◽  
Computational Aspects

We discuss computational aspects of verifying constructive criteria for ergodicity of interacting particle systems. Both discrete time (probabilistic cellular automata) and continuous time spin flip dynamics are considered. We also investigate how the criteria have to be adapted if stirring is added to the dynamics.

scholarly journals The survival of various interacting particle systems

1993 ◽  
Vol 25 (4) ◽  
pp. 1010-1012
Author(s):  
Aidan Sudbury
Keyword(s):  
Critical Phenomena ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Birth Rates ◽  
Interacting Particle

Particles may be removed from a lattice by murder, coalescence, mutual annihilation and simple death. If the particle system is not to die out, the removed particles must be replaced by births. This letter shows that coalescence can be counteracted by arbitrarily small birth-rates and contrasts this with the situations for annihilation and pure death where there are critical phenomena. The problem is unresolved for murder.

scholarly journals The survival of various interacting particle systems

1993 ◽  
Vol 25 (04) ◽  
pp. 1010-1012
Author(s):  
Aidan Sudbury
Keyword(s):  
Critical Phenomena ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Birth Rates ◽  
Interacting Particle

Particles may be removed from a lattice by murder, coalescence, mutual annihilation and simple death. If the particle system is not to die out, the removed particles must be replaced by births. This letter shows that coalescence can be counteracted by arbitrarily small birth-rates and contrasts this with the situations for annihilation and pure death where there are critical phenomena. The problem is unresolved for murder.

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