Patterns occur naturally in many physical and biological systems. By pattern, we mean a structure which has a complicated spatial dependence, but retains its shape as time passes. Prototypical examples are water waves, traveling pulses in neurons, convection cells, and tropical cyclones. This dissertation is concerned with the mathematical analysis of such patterns when they are subjected to random fluctuations in the environment, which we refer to as noise. The key questions which we address are stability, noise-induced motion, and long-time behavior of patterns.

We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space H for any s∈ [0,0∞), demonstrating persistence of regularity.