I consider a wave map equation in dimension 1+1 perturbed by a noise of co-normal type. After applying a parabolic rescaling, I show that the position converges to the solution of a deterministic heat flow that retains a memory of the noise. I then analyze the fluctuations around this deterministic limit and prove a "weak" central limit theorem. As a byproduct of this analysis, I also derive several results for the deterministic wave map and heat flow equations that appear not to be contained in the previous literature.
Parabolic rescaling of a stochastic wave map: limit and fluctuations
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