Schrödinger equation with Gaussian potential
Author:
Y. Hu
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
MathSciNet review:
3824682
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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.
References
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References
- F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Probab. Appl. (N. Y.), Springer-Verlag, London, 2008. MR 2387368
- K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger’s Equation, Grundlehren Math. Wiss., vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992
- D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. MR 1814364
- Q. Han and F. Lin, Elliptic Partial Differential Equations, 2nd ed., Courant Lect. Notes Math., vol. 1, Amer. Math. Soc., Providence, RI, 2011. MR 2777537
- T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise. An Infinite Dimensional Calculus, Mathematics and Its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1244577
- Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc. 175 (2005), no. 825. MR 2130224
- Y. Hu, Analysis on Gaussian Space, World Scientific, Singapore, 2017.
- Y. Hu, J. Huang, D. Nualart, and S. Tindel, Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency, Electron. J. Probab. 20 (2015), no. 55. MR 3354615
- Y. Hu, J. Huang, K. Le, D. Nualart, and S. Tindel, Stochastic heat equation with rough dependence in space, Ann. Probab. Probab. 45 (2017), no. 6B, 4561–4616. MR 3737918
- Y. Hu and D. Nualart, Stochastic heat equation driven by fractional noise and local time, Prob. Theory Related Fields 143 (2009), 285–328. MR 2449130
- Y. Hu, B. Øksendal, and T. Zhang, Stochastic partial differential equations driven by multiparameter fractional white noise, Stochastic Processes, Physics and Geometry: New Interplays, II (Leipzig, 1999), CMS Conf. Proc., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 327–337. MR 1803426
- Y. Hu, B. Øksendal, and T. Zhang, General fractional multiparameter white noise theory and stochastic partial differential equations, Comm. Partial Differential Equations 29 (2004), 1–23. MR 2038141
- Y. Hu and J. A. Yan, Wick calculus for nonlinear Gaussian functionals, Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 3, 399–414. MR 2506982
- Yu. G. Kondratiev, Spaces of entire functions of an infinite number of variables, connected with the rigging of a Fock space, Spectral Analysis of Differential Operators, Selecta Mathematica Sovietica 10 (1991), no. 2, 165–180. MR 1115043
- Yu. G. Kondratiev, Nuclear spaces of entire functions in problems of infinite-dimensional analysis, Soviet Math. Dokl. 22 (1980), 588–592. MR 592501
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- Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Math., vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Probab. Appl. (N. Y.), Springer-Verlag, Berlin, 2006. MR 2200233
- M. Sanz-Solé and I. Torrecilla, A fractional Poisson equation: existence, regularity and approximations of the solution, Stoch. Dyn. 9 (2009), no. 4, 519–548. MR 2589036
- L. Quer-Sardanyons and S. Tindel, Pathwise definition of second-order SDEs, Stochastic Process. Appl. 122 (2012), no. 2, 466–497. MR 2868927
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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
Keywords:
Fractional Brownian field,
fractional Gaussian noise,
Schrödinger 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