In this thesis we consider orbital stability of certain patterns in stochastic partial differential equations. We study two examples: a rotating wave in a two-dimensional reaction-diffusion equation and a soliton in a parametrically forced nonlinear Schrödinger equation. In both cases, we show that, for small noise, solutions to the stochastic equations remain close to a version of the pattern which is shifted according to some stochastic phase. We give explicit expressions for this phase, and show that it is optimal to first order in the strength of the noise.

To show stability, we construct a multiscale expansion of the solution around an arbitrarily shifted version of the pattern, and show that this expansion is accurate to second order. From this expansion an obvious candidate for the correct phase shift arises. For technical reasons we then construct a sequence of approximations to this phase shift, which is necessary to show the multiscale expansion around the correctly shifted pattern. We then combine this expansion with a deterministic stability result to get stochastic stability.

Finally, we take first steps towards formulating and proving the same results in a more general setting, where the pattern shift is represented by the action of a Lie group. We obtain some estimates necessary for the multiscale expansion, find the correct phase, and formulate necessary assumptions for the stability to hold.

In this thesis, networks of coupled quantum harmonic oscillators are studied. The dynamics of these networks are determined by single-frequency vibrations of the entire network called normal modes. We study the behavior of the nor- mal modes when the network is coupled to a thermodynamical heat bath by looking at the Lindblad Master Equation of the system. From this equation, we determine the rate at which the normal modes decay. Certain normal modes decay very slowly, and some do not decay at all. These normal modes are called quasi-noiseless and noiseless clusters respectively. We determine what happens to the noiseless clusters when the network pa- rameters are very slightly perturbed. We have found that two distinct types of noiseless clusters can be identified. The first type disappears with even the slightest perturbation, making it useless in practice. The second type instead be- comes quasi-noiseless, making it a viable candidate for applications. We show how to determine the degree to which these noiseless clusters become quasi- noiseless by looking at the other normal modes of the network. We also explain how a network of oscillators, including an optional heat bath, can be simulated with an optical setup as described in [3]. We suggest this setup can be used to verify our findings.