PARTICLE PICTURE APPROACH TO THE SELF-INTERSECTION LOCAL TIME OF BRANCHING DENSITY PROCESSES IN ${\mathcal S}' ({\mathbb R}^d)$

2007 ◽  
Vol 10 (03) ◽  
pp. 439-464 ◽  
Author(s):  
ANNA TALARCZYK
Keyword(s):  
Local Time ◽  
Particle System ◽  
Sufficient Conditions ◽  
Random Measure ◽  
Particle Systems ◽  
The Self ◽  
Intersection Local Time ◽  
Branching Particle System ◽  
Independent Particle ◽  
Density Process

The problems studied in this paper are associated with a critical branching particle system in [Formula: see text], where the particle motion is described by a Lévy process. We define the intersection local time (ILT) of two independent trees, i.e. two independent particle systems, each starting from a single particle and we give sufficient conditions for its existence. The [Formula: see text]-valued density process arises as the high density limit of a "charged" particle system, where the initial positions of particles are given by a Poisson random measure. We express the self-intersection local time of this density process by means of ILTs of pairs of trees.

DIVERGENCE RESULTS FOR SELF-INTERSECTION LOCAL TIMES OF GAUSSIAN ${\mathscr S}' ({\mathbb R}^d)$-PROCESSES

2001 ◽  
Vol 04 (04) ◽  
pp. 417-488 ◽  
Author(s):  
ANNA TALARCZYK
Keyword(s):  
Local Time ◽  
Necessary Conditions ◽  
Particle Systems ◽  
The Self ◽  
Local Times ◽  
Intersection Local Time ◽  
Convergence In Law ◽  
Stable Particle ◽  
Gaussian Formula ◽  
Intersection Local Times

For various types of Gaussian [Formula: see text]-processes we consider the case when the self-intersection local time (SILT) does not exist. We study the rate of divergence of the corresponding approximating processes obtaining, after suitable normalizations convergence in law to some [Formula: see text]-valued processes (not necessarily Gaussian). We also obtain some new necessary conditions for the existence of SILT. We give examples associated with fluctuation limits of α-stable particle systems.

Immigration processes associated with branching particle systems

1998 ◽  
Vol 30 (03) ◽  
pp. 657-675 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Particle Systems ◽  
Poisson Random Measure ◽  
General Representation ◽  
Branching Particle System ◽  
Convolution Semigroups ◽  
Entrance Law ◽  
Skew Convolution Semigroup ◽  
Special Case

The immigration processes associated with a given branching particle system are formulated by skew convolution semigroups. It is shown that every skew convolution semigroup corresponds uniquely to a locally integrable entrance law for the branching particle system. The immigration particle system may be constructed using a Poisson random measure based on a Markovian measure determined by the entrance law. In the special case where the underlying process is a minimal Brownian motion in a bounded domain, a general representation is given for locally integrable entrance laws for the branching particle system. The convergence of immigration particle systems to measure-valued immigration processes is also studied.

Immigration processes associated with branching particle systems

1998 ◽  
Vol 30 (3) ◽  
pp. 657-675 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Particle Systems ◽  
Poisson Random Measure ◽  
General Representation ◽  
Branching Particle System ◽  
Convolution Semigroups ◽  
Entrance Law ◽  
Skew Convolution Semigroup ◽  
Special Case

The immigration processes associated with a given branching particle system are formulated by skew convolution semigroups. It is shown that every skew convolution semigroup corresponds uniquely to a locally integrable entrance law for the branching particle system. The immigration particle system may be constructed using a Poisson random measure based on a Markovian measure determined by the entrance law. In the special case where the underlying process is a minimal Brownian motion in a bounded domain, a general representation is given for locally integrable entrance laws for the branching particle system. The convergence of immigration particle systems to measure-valued immigration processes is also studied.

scholarly journals A large deviation principle for a Brownian immigration particle system

2005 ◽  
Vol 42 (04) ◽  
pp. 1120-1133
Author(s):  
Mei Zhang
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Particle System ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Intensity Measure ◽  
Branching Particle System ◽  
Deviation Principle

We derive a large deviation principle for a Brownian immigration branching particle system, where the immigration is governed by a Poisson random measure with a Lebesgue intensity measure.

scholarly journals OCCUPATION TIME FLUCTUATION LIMITS OF INFINITE VARIANCE EQUILIBRIUM BRANCHING SYSTEMS

2009 ◽  
Vol 12 (04) ◽  
pp. 593-612 ◽  
Author(s):  
PIOTR MIŁOŚ
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Domain Of Attraction ◽  
Interesting Case ◽  
Occupation Time ◽  
Infinite Variance ◽  
Dependence Structure ◽  
Branching Particle System ◽  
Stable Motion ◽  
Fractional Stable Motion

We establish limit theorems for the fluctuations of the rescaled occupation time of a (d, α, β)-branching particle system. It consists of particles moving according to a symmetric α-stable motion in ℝd. The branching law is in the domain of attraction of a (1 + β)-stable law and the initial condition is the equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate α/β < d < (1 + β)α/β, critical d = (1 + β)α/β and large d > (1 + β)α/β dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments.

scholarly journals Nonexplosion of a class of semilinear equations via branching particle representations

2008 ◽  
Vol 40 (01) ◽  
pp. 250-272
Author(s):  
Santanu Chakraborty ◽  
Jose Alfredo López-Mimbela
Keyword(s):  
Differential Equation ◽  
Infinitesimal Generator ◽  
Branching Process ◽  
Random Number ◽  
Particle System ◽  
Sufficient Conditions ◽  
Multiplicative Semigroup ◽  
Branching Particle System ◽  
Semilinear Equations ◽  
Semilinear Partial Differential Equation

We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given bypk,k= 2, 3, …. The corresponding branching process is related to the semilinear partial differential equationforx∈ ℝd, whereAis the infinitesimal generator of a multiplicative semigroup and thepks,k= 2, 3, …, are nonnegative functions such thatWe obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

scholarly journals On the Self-Intersection Local Time of Subfractional Brownian Motion

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Junfeng Liu ◽  
Zhihang Peng ◽  
Donglei Tang ◽  
Yuquan Cang
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Local Time ◽  
Noise Analysis ◽  
The Self ◽  
White Noise Analysis ◽  
Intersection Local Time ◽  
Subfractional Brownian Motion

We study the problem of self-intersection local time ofd-dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis.

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