01-77 Yuri Kozitsky and Lech Wolowski

A nonlinear dynamical system on the set of Laguerre entire functions (358K, PostScrip) Feb 26, 01

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Abstract. A nonlinear modification of the Cauchy problem ${\partial} f (t, z)/{\partial t}= \theta D_z f(t,z) + zD^2_z f(t,z)$, $D_z = \partial /\partial z$, $t\in {I\!\! R}_+ = [0, +\infty)$, $z\in \hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5pt}C$, $\theta \geq 0 $, $f(0,z) = g(z)$, $g\in {\mathcal L}$ is considered. The set ${\mathcal L}$ consists of Laguerre entire functions, which one obtains as a closure of the set of polynomials having real nonpositive zeros only in the topology of uniform convergence on compact subsets of $ \ \hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5pt}C$. The modification means that the time half-line ${I\!\! R}_+$ is divided into the intervals ${\mathcal I}_n = [(n-1)\tau , n\tau ]$, $n\in {I\!\! N}$, $\tau>0$, and on each ${\mathcal I}_n $ the evolution is to be described by the above equation but at the endpoints the function $f(t, z)$ is changed: $f(n\tau , z) \rightarrow \left[ f\left(n\tau , z\delta^{-1-\lambda}\right)\right]^\delta $, with $\lambda>0$ and an integer $\delta \geq 2$. The resolving operator of such problem preserves the set ${\mathcal L}$. It is shown that for $t\rightarrow +\infty$, the asymptotic properties of $f(t, z)$ change considerably when the parameter $\tau$ reaches a threshold value $\tau_*$. The limit theorems for $\tau < \tau_* $ and for $\tau = \tau_* $ are proven. Applications, including limit theorems for weakly and strongly dependent random vectors, are given.

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