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In this thesis, we study the existence of stationary solutions for two cases. One is for random difference equations. For this, we prove the existence and uniqueness of the stationary solutions in a finite-dimensional Euclidean space Rd by applying the coupling method. The other one is for semi linear stochastic evolution equations. For this case, we follows Mohammed, Zhang and Zhao [25]'s work. In an infinite-dimensional Hilbert space H, we release the Lipschitz constant restriction by using Arzela-Ascoli compactness argument. And we also weaken the globally bounded condition for F by applying forward and backward Gronwall inequality and coupling method.

The paper considers systems of linear first-order ODEs with a randomly perturbed system matrix and stationary additive noise. For the description of the long-term behavior of such systems it is necessary to study their stationary solutions. We deal with conditions for the existence of stationary solutions as well as with their representations and the computation of their moment functions. Assuming small perturbations of the system matrix we apply perturbation techniques to find series representations of the stationary solutions and give asymptotic expansions for their first- and second-order moment functions. We illustrate the findings with a numerical example of a scalar ODE, for which the moment functions of the stationary solution still can be computed explicitly. This allows the assessment of the goodness of the approximations found from the derived asymptotic expansions.

The paper considers systems of linear first-order ODEs with a randomly perturbed system matrix and stationary additive noise. For the description of the long-term behavior of such systems it is necessary to study their stationary solutions. We deal with conditions for the existence of stationary solutions as well as with their representations and the computation of their moment functions. Assuming small perturbations of the system matrix we apply perturbation techniques to find series representations of the stationary solutions and give asymptotic expansions for their first- and second-order moment functions. We illustrate the findings with a numerical example of a scalar ODE, for which the moment functions of the stationary solution still can be computed explicitly. This allows the assessment of the goodness of the approximations found from the derived asymptotic expansions.

Internal gravity waves (GWs) are an important component of the atmospheric dynamics, significantly affecting the middle atmosphere by momentum and energy transport and deposition. In order to be able to improve global circulation models, in which the majority of the GW spectrum is not resolved, it is necessary to quantify their effects as precise as possible. We study GWs in a high-resolution simulation of the WRF model around Southern Andes, Antarctic Peninsula and South Georgia Island. We analyse a Gaussian high-pass filter method for separation of GWs from the basic flow. To overcome an observed problem of dependence of the method on a cutoff parameter, we propose an improved method that determines the parameter at each time step from the horizontal kinetic energy spectrum. The differences between the methods are further examined using the horizontal kinetic energy spectrum, vertical potential energy spectrum and forcing to the divergence equation evaluated by the active wind method, which is a recent theory-based method that divides the flow into a balanced flow and a perturbation field. The results suggest that the high-pass filter method does not produce correct results for time periods with strong wave activity.