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...

We consider two exit problems for the Korteweg-de Vries equationperturbed by an additive white in time and colored in space noise of amplitude. The initial datum gives rise to a soliton when = 0. It has been provedrecently that the solution remains in a neighborhood of a randomly...

Uniform large deviations for the laws of the paths of the solutionsof the stochastic nonlinear Schr¨odinger equation when the noise converges tozero are presented. The noise is a real multiplicative Gaussian noise. It iswhite in time and colored in space. The path space considered allows...

Exit from a neighborhood of zero for weakly damped stochasticnonlinear SchrÄodinger equations is studied. The small noise is either complexand of additive type or real and of multiplicative type. It is white in time andcolored in space. The neighborhood is either in L2 or in H1. The small...

Exit from a neighborhood of zero for weakly damped stochasticnonlinear SchrÄodinger equations is studied. The small noise is either complexand of additive type or real and of multiplicative type. It is white in time andcolored in space. The neighborhood is either in L2 or in H1. The small...

Sample path large deviations for the laws of the solutions of sto-chastic nonlinear SchrÄodinger equations when the noise converges to zero arepresented. The noise is a complex additive gaussian noise. It is white in timeand colored space wise. The solutions may be global or blow-up in ¯nite...

We propose a new method of estimation in high-dimensional linear regression model. It allows for very weak distributional assumptions including heteroscedasticity, and does not require the knowledge of the variance of random errors. The method is based on linear programming only, so that its...