Stochastic nonlinear Schrödinger equations driven by a fractional noise, well-posedness, large deviations and support. (English) Zbl 1127.60057

Gautier, Eric

Electron. J. Probab. 12, 848-861 (2007).

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 \(H>1/2\).

MSC:

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

Keywords:

fractional Brownian motion; local well-posedness; Kerr nonlinearities