Analytic methods in the theory of quadric stochastic operators
type: Item
instance of: scholarly article
title: Analytic methods in the theory of quadric stochastic operators
zbMATH Open document ID: 0696.60062
DOI: 10.1007/BF01063328
author: T. A. Sarymsakov
author: Nasir N. Ganikhodjaev
published in: Journal of Theoretical Probability
publication date: 1990-01-01T00:00:00Z
review text: The aim of the paper is to obtain equations satisfied by the transition probabilities or the transition probability densities when quadric stochastic processes are considered in continuous time. The authors assume that there is an ``interaction time'' or ``time lag'' and study transitions occurring at instants separated by intervals longer than the time lag. But no hypothesis is made as to what happens during shorter intervals of time. Or am I missing something? Also, I could not reproduce some of their computations, but that could be my fault! Consider their \[ (2.3)\quad P_{ij,k}P_{ab,i}P_{lm,j}x_ ax_ bx_{\ell}x_ m=P_{ab,k}^{[0,2]}x_ ax_ b \] (with Einstein's summation convention). Differentiating both sides with respect to \(x_ a\) and \(x_ b\) I obtain \[ P_{ab,k}^{[0,2]}=2P_{ij,k}P_{ab,i}P_{lm,j}x_{\ell}x_ m+4P_{ij,k}P_{al,i}P_{bm,j}x_{\ell}x_ m \] where the symmetry assumptions \(P_{ij,k}=P_{ji,k}\) are used. Compare with their list I- VI.
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60G12
zbMATH DE Number: 4140953
zbMATH Keywords: quadric stochastic processes
zbMATH Keywords: interaction time
reviewed by: Henryk Gzyl
MaRDI profile type: Publication
cites work: Quadratic transformations: a model for population growth. I
description: scientific article; zbMATH DE number 4140953