Parabolic scaling of a stochastic wave map with co-normal noise: limit and fluctuations

This paper investigates the parabolic scaling limit of a damped stochastic wave map from the real line into the two-dimensional sphere, perturbed by multiplicative Gaussian noise of co-normal type. We prove that under this rescaling, the solutions converge to those of the deterministic heat flow for harmonic maps, revealing a transition from stochastic hyperbolic to deterministic parabolic dynamics. We further analyze the fluctuations around this limit, proving a weak central limit theorem and identifying the limiting process as the solution to a linear stochastic partial differential equation. The study combines tools from geometric analysis, stochastic calculus, and functional analysis, offering insights into the interplay between geometry, noise, and scaling in nonlinear stochastic systems.