Space scaling limit theorems for infinite particle branching brownian motions with immigration

2005 ◽  
pp. 91-99
Author(s):  
Luis G. Gorostiza
Keyword(s):  
Limit Theorems ◽  
Scaling Limit ◽  
Brownian Motions ◽  
Infinite Particle

On Limit Theorems for Brownian Motions on Unbounded Fractal Sets

2000 ◽  
pp. 227-237
Author(s):  
Masatoshi Fukushima
Keyword(s):  
Limit Theorems ◽  
Brownian Motions ◽  
Fractal Sets

Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

2021 ◽  
Vol 34 (2) ◽  
pp. 141-173
Author(s):  
Hirofumi Osada
Keyword(s):  
Random Matrices ◽  
Point Processes ◽  
Stochastic Dynamics ◽  
Particle System ◽  
Infinite Dimensions ◽  
Algebraic Construction ◽  
Brownian Motions ◽  
Content Type ◽  
Infinite Dimensional ◽  
Infinite Particle

We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

Limit theorems for infinite particle systems

1971 ◽  
Vol 20 (2) ◽  
pp. 87-101 ◽  
Author(s):  
N. A. Weiss
Keyword(s):  
Limit Theorems ◽  
Particle Systems ◽  
Infinite Particle Systems ◽  
Infinite Particle

scholarly journals THE FULL BROWNIAN WEB AS SCALING LIMIT OF STOCHASTIC FLOWS

2006 ◽  
Vol 06 (02) ◽  
pp. 213-228
Author(s):  
LUIZ RENATO FONTES ◽  
CHARLES M. NEWMAN
Keyword(s):  
Scaling Limit ◽  
Stochastic Flows ◽  
Convergence Theorems ◽  
Brownian Motions ◽  
One Dimensional ◽  
General Characterization

In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all spacetime trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.

NONCOMMUTATIVE BROWNIAN MOTIONS INDEXED BY PARTIALLY ORDERED SETS

2010 ◽  
Vol 13 (04) ◽  
pp. 629-661 ◽  
Author(s):  
ANNA KULA ◽  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorems ◽  
Ordered Set ◽  
Positive Cone ◽  
Symmetric Matrices ◽  
Ordered Sets ◽  
Brownian Motions ◽  
Partially Ordered ◽  
Independent Increments ◽  
Lorentz Cone ◽  
The Given

We construct noncommutative Brownian motions indexed by partially ordered subsets of Euclidean spaces. The noncommutative independence under consideration is the bm-independence and the time parameter is taken from a positive cone in a vector space ([Formula: see text], the Lorentz cone or the positive definite real symmetric matrices). The construction extends the Muraki's idea of monotonic Brownian motion. We show that our Brownian motions have bm-independent increments for bm-ordered intervals. The appropriate version of the Donsker Invariance Principle is also proved for each positive cone. It requires the bm-Central Limit Theorems related to intervals in the given partially ordered set of indices.

scholarly journals Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions

1978 ◽  
Vol 14 (3) ◽  
pp. 741-788 ◽  
Author(s):  
Richard A. Holley ◽  
Daniel W. Stroock
Keyword(s):  
Brownian Motions ◽  
Infinite Particle

An infinite particle system with continual input and random death

1978 ◽  
Vol 10 (04) ◽  
pp. 764-787
Author(s):  
J. N. McDonald ◽  
N. A. Weiss
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Limit Theorems ◽  
Particle System ◽  
Countable State Space ◽  
Infinite Particle System ◽  
Large Numbers ◽  
Countable State ◽  
Infinite Particle ◽  
Number Of Particles

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

scholarly journals Scaling limit of two-component interacting Brownian motions

2018 ◽  
Vol 46 (4) ◽  
pp. 2038-2063 ◽  
Author(s):  
Insuk Seo
Keyword(s):  
Scaling Limit ◽  
Brownian Motions ◽  
Two Component

Functionals of an infinite particle system with independent motions, creation, and annihilation

1974 ◽  
Vol 6 (4) ◽  
pp. 636-650 ◽  
Author(s):  
P. A. Jacobs
Keyword(s):  
State Space ◽  
Central Limit ◽  
Limit Theorems ◽  
Particle System ◽  
Central Limit Theorems ◽  
The Other ◽  
Strong Laws ◽  
Infinite Particle System ◽  
Infinite Particle ◽  
Random Times

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained.

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