Abstract
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions.
Highlights
Our aim is to study the longtime dynamics of stochastic evolution equations using an approach that is different from the classical one
We consider parabolic problems with random differential operators and use a pathwise representation formula to show that the solution operator generates a random dynamical system and to prove that it possesses random attractors of finite fractal dimension
Our aim is to show that problem (1.1) generates a random dynamical system using the concept of pathwise mild solutions and to prove the existence of random attractors
Summary
Our aim is to study the longtime dynamics of stochastic evolution equations using an approach that is different from the classical one. We consider parabolic problems with random differential operators and use a pathwise representation formula to show that the solution operator generates a random dynamical system and to prove that it possesses random attractors of finite fractal dimension. Mathematics Subject Classification: 60H15, 37H05, 37L55 Keywords: Stochastic parabolic evolution equations, Pathwise mild solution, Random attractors, Fractal dimension. Our aim is to show that problem (1.1) generates a random dynamical system using the concept of pathwise mild solutions and to prove the existence of random attractors. 2, we collect basic notions and results from the theory of random dynamical systems and nonautonomous stochastic evolution equations and recall an existence result for random exponential attractors. Our paper provides a first, simple example that illustrates how the concept of pathwise mild solutions can be used to show the existence of global and exponential random attractors for SPDEs with random differential operators. Another interesting aspect would be to investigate the regularity of random attractors
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