Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay (Q854716)

The authors consider an ordinary delay differential equation (DDE) of the form \[ {d\over dt}\,x(t)=F(x(t))+G(x(t-\tau(t,\omega))) \] with bounded random delay \(\tau(t,\omega)=\tau(\theta_t\omega)\in[\tau_*,\tau^*]\) on a probability space \((\Omega, {\mathcal F},{\mathbf P})\) and its split implicit Euler discretization scheme (SIES) \[ x_{n+1}=x_n+F(x_{n+1})\Delta +G(x_{n-[\tau(\theta_{n\Delta}\omega)/\Delta]})\Delta, \] where \(\theta\) is a metric dynamical system on \(\Omega\) representing the noise. Under certain assumptions, it is proved that both the DDE and SIES generate respectively continuous- and discrete-time random dynamical systems both having stochastic stationary solutions that pathwise attract all other trajectories of the corresponding equations. Moreover, the stochastic stationary solution of SIES converges to that of the DDE as the stepsize \(\Delta\to0\).