Minimal jointly uniform attractor for nonautonomous random dynamical systems
Pedro Catuogno, Alexandre do Nascimento Oliveira-Sousa, Paulo Ruffino
TL;DR
This work introduces the minimal jointly uniform attractor (MJUA) for nonautonomous random dynamical systems, capturing long-time attraction that depends jointly on time and random parameters. It develops existence results under uniform absorbing conditions and proves stability via compactification of the symbol space, linking NRDS to deterministic hull dynamics through a skew-product framework. The theory is extended to random dynamical systems and applied to stochastic differential equations, establishing a random cocycle conjugacy that yields explicit attractors in examples. Together, the results provide a cohesive framework for analyzing attractors in NRDS with time-dependent noise and random influences.
Abstract
We introduce a notion of minimal uniform attractor for nonautonomous random dynamical systems, which depends jointly on time and on a random parameter. Several examples are provided to illustrate the concept and to compare it with existing notions of uniform attractors in the literature. We further apply the abstract theory to nonautonomous random differential equations with a non-compact symbol space. In particular, we develop a method to compactify the symbol space, by adapting techniques from the theory of deterministic nonautonomous differential equations. We also establish the stability of the minimal jointly uniform attractor by exploiting the relationship between deterministic and random dynamics. Finally, we show that such structures arise naturally in stochastic differential equations whose noise terms carry additional time dependence, by establishing a topological conjugacy between the resulting stochastic flows and suitable random differential equations.