| This thesis is divided into two major parts. First we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise. The stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system. However, the stability of the second moment is altered by the perturbation. We obtain, using Laplace transform techniques, necessary and sufficient conditions for the second moment to be bounded. Then we establish the stability criteria for stochastic differential equations with Markovian switching using the comparison principle. These criteria include stability in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Next, we study the uniqueness of the invariant measure for the wave equation with noise. We will use a coupling technique and others from the theory of Markov chains on general state spaces. The application of these Markov chain results leads to straightforward proofs of ergodicity of SDEs. The key points which need to be verified are the existence of a Lyapunov function including returns to a compact set, a uniformly reachable point from within that set and some smoothness of the probability densities. |
