TY - JOUR
T1 - Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schrödinger equations
AU - Forlano, Justin
AU - Seong, Kihoon
N1 - Funding Information:
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 and Tadahiro Oh’s ERC starting grant no. 637995 “ProbDynDispEq”. K.S. was partially supported by National Research Foundation of Korea (grant NRF-2019R1A5A1028324). The authors would like to kindly thank Tadahiro Oh and Nikolay Tzvetkov for suggesting the problem, for their continued support and for informing us that the -integrability assumption in [38, Corollary 1.4] can be weakened to -integrability. The authors are also grateful to Nikolay Tzvetkov for suggesting the application of Proposition 1.10 to the Lp -stability result in Proposition 1.11. The authors also wish to thank the anonymous referees for their helpful comments.
Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.
PY - 2022
Y1 - 2022
N2 - We study the transport property of Gaussian measures on Sobolev spaces of periodic functions under the dynamics of the one-dimensional cubic fractional nonlinear Schrödinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures, following the approach by Debussche and Tsutsumi (2021). Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and, as a corollary, a formula for the two-point function arising in wave turbulence theory. We also obtain improved regularity results in the weakly dispersive case, extending those by the first author and Trenberth (2019). Our proof combines the approach introduced by Planchon, Tzvetkov and Visciglia (2020) and that of Debussche and Tsutsumi (2021).
AB - We study the transport property of Gaussian measures on Sobolev spaces of periodic functions under the dynamics of the one-dimensional cubic fractional nonlinear Schrödinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures, following the approach by Debussche and Tsutsumi (2021). Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and, as a corollary, a formula for the two-point function arising in wave turbulence theory. We also obtain improved regularity results in the weakly dispersive case, extending those by the first author and Trenberth (2019). Our proof combines the approach introduced by Planchon, Tzvetkov and Visciglia (2020) and that of Debussche and Tsutsumi (2021).
KW - Fractional nonlinear Schrödinger equation
KW - Gaussian measure
KW - quasi-invariance
UR - http://www.scopus.com/inward/record.url?scp=85129127051&partnerID=8YFLogxK
U2 - 10.1080/03605302.2022.2053861
DO - 10.1080/03605302.2022.2053861
M3 - Article
AN - SCOPUS:85129127051
SN - 0360-5302
VL - 47
SP - 1296
EP - 1337
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
IS - 6
ER -