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Theory of Probability and Mathematical Statistics

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Schrödinger equation with Gaussian potential


Author: Y. Hu
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal: Theor. Probability and Math. Statist. 98 (2019), 115-126
MSC (2010): Primary 60G15, 60G22, 46F25
DOI: https://doi.org/10.1090/tpms/1066
Published electronically: August 19, 2019
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Abstract: This paper studies the Schrödinger equation with fractional Gaussian noise potential of the form $ \Delta u(x)= u(x)\diamond \dot W(x)$, $ x\in D$, $ u(x)= \phi (x)$, $ x\in \partial D$, where $ \Delta $ is the Laplacian on the $ d$-dimensional Euclidean space $ \mathbb{R}^d$, $ D\subseteq \mathbb{R}^d$ is a given domain with smooth boundary $ \partial D$, $ \phi $ is a given nice function on the boundary  $ \partial D$, and $ \dot W$ is the fractional Gaussian noise of Hurst parameters $ (H_1, \ldots , H_d)$ and $ \diamond $ denotes the Wick product. We find a family of distribution spaces $ (\mathbb{W}_{\lambda }, {\lambda }>0)$, with the property $ \mathbb{W}_{{\lambda }}\subseteq \mathbb{W}_\mu $ when $ {\lambda }\le \mu $, such that under the condition $ \sum _{i=1}^d H_i>d-2$, the solution exists uniquely in $ \mathbb{W}_{{\lambda }_0} $ when $ {\lambda }_0$ is sufficiently large and the solution is not in $ \mathbb{W}_{{\lambda }_1}$ when $ {\lambda }_1$ is sufficiently small.


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Additional Information

Y. Hu
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta at Edmonton, Edmonton, T6G 2G1, Canada
Email: yaozhong@ualberta.ca

DOI: https://doi.org/10.1090/tpms/1066
Keywords: Fractional Brownian field, fractional Gaussian noise, Schr\"odinger equation, distribution spaces, chaos expansion, Poisson equation, multiplicative noise
Received by editor(s): February 1, 2018
Published electronically: August 19, 2019
Article copyright: © Copyright 2019 American Mathematical Society