scholarly journals Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field

2013 ◽  
Vol 123 (3) ◽  
pp. 813-838 ◽  
Author(s):  
Hirofumi Osada
Keyword(s):  
Random Point ◽  
Infinite Dimensions ◽  
Interaction Potentials ◽  
Brownian Motions ◽  
Logarithmic Interaction ◽  
Point Field

scholarly journals Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions

2020 ◽  
Author(s):  
Yosuke Kawamoto ◽  
Hirofumi Osada ◽  
Hideki Tanemura
Keyword(s):  
Finite Volume ◽  
Stochastic Dynamics ◽  
Matrix Theory ◽  
Dirichlet Forms ◽  
Interaction Potentials ◽  
Brownian Motions ◽  
Infinite Systems ◽  
Point Field ◽  
First Main Theorem ◽  
Class Interaction

Abstract The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms $ (\mathcal {E}^{\mathsf {upr}},\mathcal {D}^{\mathsf {upr}})$ ( E u p r , D u p r ) and $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of $ {C_{0}^{3}} $ C 0 3 -class.

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.

scholarly journals Cauchy–Laguerre Two-Matrix Model and the Meijer-G Random Point Field

2013 ◽  
Vol 326 (1) ◽  
pp. 111-144 ◽  
Author(s):  
M. Bertola ◽  
M. Gekhtman ◽  
J. Szmigielski
Keyword(s):  
Matrix Model ◽  
Random Point ◽  
Point Field

Method of features construction for remote sensing images based on the characteristics of random point fields

2017 ◽  
Vol 2017 (45) ◽  
pp. 90-95
Author(s):  
R.Ya. Kosarevych ◽  
◽  
O.A. Lutsyk ◽  
B.P. Rusyn ◽  
V.V. Korniy ◽  
...  
Keyword(s):  
Remote Sensing ◽  
Dynamic Range ◽  
Texture Features ◽  
Input Image ◽  
Image Resolution ◽  
Random Point ◽  
Local Maxima ◽  
Image Fragment ◽  
Point Field ◽  
Field Element

Texture features are widely used in remote sensing image classification. In most cases they are extracted from grayscale images without taking color information into consideration. The texture descriptors, which consist of characteristics of random point fields formed for pixels of distinct intensity of grayscale and color band images are presented. The input image is divided into fragments for the elements of each of which the histogram is constructed and their local maxima are determined. Size of fragments are chosen depending on image resolution. For each of the intensity of the dynamic range of the image, a random point field, as a set of geometric centers of fragments, is formed. By the formed configuration, each field is classified as cluster, regular or random. To form a description of image elements a distribution of the number of field elements for each intensity and fragment is constructed. Separately, the vectors of the point field element for each intensity in the image fragment and the point field element for the selected intensity are formed. Experimental results demonstrate that proposed descriptors yield performance compared to other state-of-the-art texture features.

On a Quantum Model of Brain Activities — Distribution of the Outcomes of EEG-Measurements and Random Point Fields

2012 ◽  
Vol 19 (04) ◽  
pp. 1250025 ◽  
Author(s):  
Karl-Heinz Fichtner ◽  
Kei Inoue ◽  
Masanori Ohya
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Probability Distributions ◽  
Quantum States ◽  
Random Point ◽  
Quantum Model ◽  
Classical Probability ◽  
Point Configurations ◽  
Point Field ◽  
The Brain

Considering models based on classical probability theory, states of signals in the brain should be identified with probability distributions of certain random point fields representing the configuration of excited neurons. Then the outcomes of EEG-measurements can be considered as random variables being certain functions of that random point field. In practice, specialists use certain statistical methods evaluating the outcomes of the sequence of these measurements. To make these statistical investigations precise, one should know the distribution of the stochastic process on the space of point configurations representing the time evolution of the configuration of excited neurons in the brain. Up to now that distribution is totally unknown. In this paper we consider time evolutions of random point fields as well as the distribution of the outcomes of EEG-measurements related to unitary evolutions of certain quantum states used in [4, 5, 10 – 14] in order to describe activities of the brain.

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