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Stochastic nonlinear Schrödinger equations driven by a fractional noise, well-posedness, large deviations and support. (English) Zbl 1127.60057
Summary: We consider stochastic nonlinear Schrödinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter \(H\) in (0,1) and colored in space with a nuclear space correlation operator. We study local well-posedness. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove sample path large deviations and support results in a space of Hölder continuous in time until blow-up paths. We consider Kerr nonlinearities when \(H>1/2\).

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F10 Large deviations
35Q55 NLS equations (nonlinear Schrödinger equations)
35R60 PDEs with randomness, stochastic partial differential equations
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