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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
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 
. 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。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
. 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
。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。
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