Long Time Behavior of Stochastic NLS with a Small Multiplicative Noise

We prove the global space-time bound for the mass critical nonlinear Schrödinger equation perturbed by a small multiplicative noise in dimension three. We also prove a global Strichartz space-time bound for the linear stochastic model, which is new itself and serves a prototype model for the nonlinear case. The proof combines techniques from [FX21a], [FZ20] as well as local smoothing estimates for linear Schrödinger operators.

[1]  A. Debussche,et al.  A Stochastic Nonlinear Schrödinger Equation¶with Multiplicative Noise , 1999 .

[2]  F. Hornung.  The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates , 2016, 1611.07325.

[3]  B. Dodson.  Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 , 2010, 1010.0040.

[4]  T. Tao.  Nonlinear dispersive equations : local and global analysis , 2006 .

[5]  A. Soffer,et al.  Decay estimates for Schrödinger operators , 1991 .

[6]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1919, Nature.

[7]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .

[8]  Michael Röckner,et al.  Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach , 2014, Journal of Nonlinear Science.

[9]  D. Burkholder.  Distribution Function Inequalities for Martingales , 1973 .

[10]  Luis Vega.  Schrödinger equations: pointwise convergence to the initial data , 1988 .

[11]  M. Röckner,et al.  Scattering for Stochastic Nonlinear Schrödinger Equations , 2018, Communications in Mathematical Physics.

[12]  Zehua Zhao,et al.  On long time behavior for stochastic nonlinear Schr\"odinger equations with a multiplicative noise. , 2020, 2010.11045.

[13]  T. Tao,et al.  Endpoint Strichartz estimates , 1998 .

[14]  B. Davis,et al.  Integral Inequalities for Convex Functions of Operators on Martingales , 2011 .

[15]  Jean Bourgain,et al.  Mathematical aspects of nonlinear dispersive equations , 2007 .

[16]  Xue Ping Wang.  Time-decay of semigroups generated by dissipative Schrödinger operators , 2012 .

[17]  P. Sjölin,et al.  Regularity of solutions to the Schrödinger equation , 1987 .

[18]  Weijun Xu,et al.  A Wong-Zakai theorem for mass critical NLS , 2019, 1906.06616.

[19]  A. Bourgougnon.  Comptes Rendus de l'Académie des Sciences. , 1879 .

[20]  Wei Liu,et al.  The stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise , 2020, 2001.05259.

[21]  J. Saut,et al.  Effets régularisants locaux pour des équations dispersives générales , 1987 .

[22]  Deng Zhang.  Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases , 2018, The Annals of Applied Probability.

[23]  Szymon Peszat,et al.  Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process , 1999 .

[24]  Weijun Xu,et al.  Decay of the stochastic linear Schrödinger equation in $$d \ge 3$$ d ≥ 3 with small multiplicative noise , 2018, Stochastics and Partial Differential Equations: Analysis and Computations.

[25]  Z. Brzeźniak.  On stochastic convolution in banach spaces and applications , 1997 .

[26]  B. M. Fulk.  MATH , 1992 .

[27]  B. Dodson.  Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 , 2016 .

[28]  A. Millet,et al.  On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold , 2012, 1209.3578.

[29]  Christopher Smith,et al.  Volume 10 , 2021, Engineering Project Organization Journal.

[30]  A. Debussche,et al.  The Stochastic Nonlinear Schrödinger Equation in H 1 , 2003 .

[31]  A. Kiselev,et al.  Maximal Functions Associated to Filtrations , 2001 .

[32]  American Journal of Mathematics , 1886, Nature.