Analytic methods in the theory of quadric stochastic operators
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.
scientific article; zbMATH DE number 4140953