scholarly journals Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.

2007 ◽  
Vol 12 (0) ◽  
pp. 862-887 ◽  
Author(s):  
Ben Hambly ◽  
Liza Jones
Keyword(s):  
Stable Processes ◽  
Brownian Motions ◽  
Infinite Systems ◽  
Alpha Stable Processes

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.

A novel adaptive lattice filtering algorithm for alpha-stable processes

2008 ◽  
Vol 18 (6) ◽  
pp. 977-984
Author(s):  
Zhijin Zhao ◽  
Junna Shang ◽  
Kehai Dong
Keyword(s):  
Stable Processes ◽  
Filtering Algorithm ◽  
Alpha Stable Processes

Least L/sub p/-norm estimation of autoregressive model coefficients of symmetric /spl alpha/-stable processes

1997 ◽  
Vol 4 (7) ◽  
pp. 201-203 ◽  
Author(s):  
E.E. Kuruoglu ◽  
P.J.W. Rayner ◽  
W.J. Fitzgerald
Keyword(s):  
Autoregressive Model ◽  
Stable Processes ◽  
Alpha Stable Processes
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