Correlation effects for stochastic zeros of Sturm-Liouville equations
Consider the stochastic Sturm-Liouville equations of the form \[ (1)\quad (p(x,\omega)u')'+q(x,\omega)u=0,\quad (x,\omega)\in [0,\infty)\times \Omega,\quad u(0,\omega)=0 \] with probability 1, and the most obvious deterministic approximation to such, the classical Sturm-Liouville equation \[ (2)\quad (P(x)v')'+Q(x)v=0,\quad x\in [0,\infty),\quad v(0)=0, \] where P and Q are the expected values of p and q relative to a given probability space \(\Omega\). Denote the smallest positive zeros of such solutions of (1) and (2) by \(\xi\) (\(\omega)\) and \(\eta\), respectively, and by X the expected value of \(\xi\) (\(\omega)\) relative to \(\Omega\). The paper focuses on the nature of the correlation between p and q in establishing criteria for \(X\geq \eta\).
scientific article; zbMATH DE number 4134466
scientific article