Necessary and sufficient conditions for doubly stochasticity of infinite-dimensional quadratic operators
A quadratic stochastic operator (q.s.o.\ for short) is a map \(V:S^{m-1}\to S^{m-1}\), defined over the \((m-1)\)-dimensional simplex \[ S^{m-1}=\big\{x=(x_1,\dots,x_m)\in \mathbb{R}^m : x_i\geq0\;\text{for all \(i\)},\;\sum_{i=1}^mx_i=1\big\}, \] given by \((Vx)_k=\sum_{i,j=1}^mp_{ij k}x_ix_j\), \(k=1,2,\dots,m\), where the coefficients \(p_{ij k}\) satisfy \(p_{ij k}=p_{ji k}\geq 0 \) and \(\sum_{k=1}^m p_{ij k}=1\). The interest in quadratic stochastic operators arises from their relationship with many different physical and genetic dynamical systems. A particular type of q.s.o.\ is that of doubly stochastic q.s.o., defined as those satisfying \(Vx \prec x\), where \( \prec\) denotes a classical majorization. In this paper, the author deals with the extension of the notion of q.s.o.\ to the infinite-dimensional setting. Thus, the infinite-dimensional simplex \[ S =\big\{x=(x_1,x_2,\dots)\in \ell_1 : x_i\geq0\;\text{for all \(i\)},\;\sum_{i=1}^\infty x_i=1\big\} \] is considered and then an infinite-dimensional q.s.o.\ is defined to be a map \(V:S\to S\) given by \((Vx)_k=\sum_{i,j=1}^\infty a_{ij k}x_ix_j\) for all \(k=1,2,\dots\), where the coefficients \(a_{ij k}\) are such that \(a_{ij k}=a_{ji k}\geq 0 \) and \(\sum_{k=1}^\infty a_{ij k}=1\). By extending the majorization to the elements of the infinite-dimensional simplex \(S\), the author also introduces the notion of doubly stochasticity for infinite-dimensional q.s.o. Indeed, for any \(x=(x_1,x_2,\dots)\in S\), and denoting by \(x_\downarrow=(x_{[1]},x_{[2]},\dots)\) the nonincreasing rearrangement of the entries of \(x\), that is, \(x_{[1]}\geq x_{[2]}\geq \dots\), it is said that \(y\) majorizes \(x\) (or that \(x\) is majorized by \(y\)) if \( x\prec y\) for any \(x,y\in S\). Then an infinite-dimensional q.s.o.\ is said to be doubly stochastic if \(Vx\prec x\) for all \(x\in S\). Contrary to the finite-dimensional setting, where doubly stochastic operators are proved to satisfy ergodicity conditions, in the infinite-dimensional case, similar ergodicity properties are easily checked to fail. The paper is then focused on providing a characterization of infinite-dimensional doubly stochastic q.s.o. Such a characterization is given matricially in terms of (infinite) symmetric matrices which are solutions of the matrix equation \(A=\frac{T+T'}{2}\) into a class of (infinite) matrices \(T\), extending the notion of square substochastic matrix.
scientific article; zbMATH DE number 6127327
scientific article