Cooperative random and stochastic differential equations
The authors provide a systematic study of order-preserving (or monotone) random dynamical systems generated by cooperative random differential equations (RDEs) or stochastic differential equations (SDEs). The RDEs considered are pathwise ordinary differential equations in \(\mathbb{R}^d_+\) of the form \[ \dot x(t)=f\big(\theta_t\omega,x(t)\big), \] where \(\theta=(\Omega,{\mathcal F},{\mathbf P},\{\theta_t\), \(t\in \mathbb{R}\}\) is a metric dynamical system. The SDEs considered are Stratonovich equation systems in \(\mathbb{R}^d_+\) of the form \[ dx_i=f_i(x_1,\ldots,x_d) dt+\sum_{j=1}^m\sigma_{ij}(x_i)\circ dW_t^j,\quad i=1,\ldots,d, \] where \(W\) is an \(m\)-dimensional Wiener process with two-sided time \(R\). The main results of the paper concern the long-term behavior of these systems, in particular, the existence of equilibria and attractors and a limit trichotomy theorem. Several applications (models of control of the protein synthesis in a cell, of gonorrhea infection, and of symbiotic interaction in a random environment) are treated in detail.
order-preserving or monotone random dynamical system
scientific article; zbMATH DE number 1882960