Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise

Abstract

We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the "tails" of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.2010 Mathematics Subject Classification. 37L55; 35B40; 60H15. Key words and phrases. Random exponential attractor, fractal dimension, multiplicative white noise, discrete long wave-short wave resonance equation.

Statistical solutions and Kolmogorov entropy for the lattice long-wave–short-wave resonance equations in weighted space

“…For example, the global attractors, exponential attractors and their fractal dimension, pullback attractor, uniform attractors for the first-order and second-order LDSs were investigated in [25,[39][40][41]; [42] researched the random exponential attractor for the stochastic LDSs; the uniform global attractor, pullback attractor and random attractor in weighted space were studied in [1,14,15]. For the lattice long-wave-short-wave resonance equations (1.1)-(1.2), the existence of the attractors and kernel sections was verified in [20,23,28]; the existence of invariant Borel measures was established in [26]. However, to the best of our knowledge, there is no reference investigating the statistical solution of equations (1.1)-(1.2) in weighted space.…”