Stochastic Nonlinear Schrödinger Equations Driven by a Fractional Noise Well Posedness, Large Deviations and Support

We consider stochastic nonlinear Schr¨odinger equations driven byan additive noise. The noise is fractional in time with Hurst parameter H in(0, 1). It is also colored in space and the space correlation operator is assumed tobe nuclear. We study the local well-posedness of the equation. Under adequateassumptions on the initial data, the space correlations of the noise and for somesaturated nonlinearities, we prove a sample path large deviations principle and asupport result. These results are stated in a space of exploding paths which areH¨older continuous in time until blow-up. We treat the case of Kerr nonlinearitieswhen H > 12 .