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STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION WITH ALMOST SPACE–TIME WHITE NOISE

Part of: Equations of mathematical physics and other areas of application Stochastic analysis

Published online by Cambridge University Press:  21 June 2019

JUSTIN FORLANO ,
TADAHIRO OH  and
YUZHAO WANG
JUSTIN FORLANO
Affiliation:
Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK email j.a.forlano@sms.ed.ac.uk
TADAHIRO OH*
Affiliation:
School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK email hiro.oh@ed.ac.uk
YUZHAO WANG
Affiliation:
School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham, B15 2TT, UK email y.wang.14@bham.ac.uk
*
✉hiro.oh@ed.ac.uk
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Abstract

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

Keywords

stochastic nonlinear Schrödinger equationwell-posednesswhite noiseFourier–Lebesgue spaces

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 44 - 67
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

J. F. was supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. T. O. was supported by the European Research Council (grant no. 637995 ‘ProbDynDispEq’).

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