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A nonlinear Schrödinger equation with fractional noise
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2021-03-30 , DOI: 10.1090/tran/8368
Aurélien Deya , Nicolas Schaeffer , Laurent Thomann

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.


中文翻译:

具有分数噪声的非线性Schrödinger方程

摘要:我们研究了在空间维度上具有二次非线性和时空分数摄动的随机Schrödinger方程$ d \ leq 3 $。当赫斯特指数足够大时,我们使用经典参数证明问题的局部适定性。但是,对于较小的赫斯特指数,即使对方程式的解释也需要一定注意。在这种情况下,必须将重新规范化过程纳入其中,从而导致对该模型进行Wick类型的解释。然后,我们的定点参数涉及Schrödinger组的一些特定正则化属性,这使我们能够应对解决方案的强烈不规则性。
更新日期:2021-04-30
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