A nonlinear Schrödinger equation with fractional noise

Deya, Aurélien; Schaeffer, Nicolas; Thomann, Laurent

doi:10.1090/tran/8368

A nonlinear Schrödinger equation with fractional noise
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by Aurélien Deya, Nicolas Schaeffer and Laurent Thomann PDF

Trans. Amer. Math. Soc. 374 (2021), 4375-4422 Request permission

Abstract:

We study a stochastic Schrödinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension $d\leq 3$. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schrödinger group, which allows us to cope with the strong irregularity of the solution.

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