Probabilistic study of a dynamical system.
The paper investigates relations between a nonlinear dynamical system on~\(\mathbb C^2\) and the long term behaviour of a certain branching process. More specifically, the dynamical system is given by \(\dot z=(R+Q)z-2z^2\) with \(z\in\mathbb C^2\), where~\(R\) is a diagonal matrix with positive entries, \(Q\) has positive off-diagonal elements and vanishing row sums, and~\(z^2\) is to be understood component-wise. The flow \((\varphi_t)\) induced by this system on \(\{(z_1,z_2)\in\mathbb C^2:\text{Re}\,z_i\geq0\}\) is shown to satisfy \( e^{-x(\varphi_tz)}=E_x e^{-X_t\,z}\), where \((X_t)\) is the solution of a stochastic differential equation in~\(\mathbb R^2\) with linear drift coefficient~\(R+Q\) and diffusion coefficient the component-wise square roots of~\(X\) with a two-dimensional Wiener process, and~\(E_x\) denotes expectation for the law induced by starting~\((X_t)\) in the initial value~\(x\). The process~\((X_t)\) is argued to be best interpreted as a measure valued process on~\(I=\{1,2\}\). Several further results, relating properties of the dynamical system~\((\varphi_t)\) to those of~\((X_t)\), are derived, amongst them global convergence to the attractor and assertions in the realm of Poincaré-Dulac theory.
scientific article; zbMATH DE number 1696293