Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise

Abstract

<p style='text-indent:20px;'>We mainly consider the existence of random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) defined on the space of infinite sequences with complex-valued components. Then we transfer the considered stochastic Schrödinger lattice system into a random system with random parameters and quasi-periodic forces without white noise through Ornstein-Uhlenbeck process, whose solutions generate a jointly continuous NRDS. Thirdly, we prove the existence of a uniform absorbing random set for this NRDS. Fourthly, we construct a bounded closed random set and verify the Lipschitz continuity of the NRDS on this random set, and next we decompose the difference between two solutions into a sum of two parts including some random variables with bounded expectation. Finally, we obtain the existence of a random uniform exponential attractor for the considered system.</p>