STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR STOCHASTIC DISPERSIVE EQUATIONS WITH LINEAR MULTIPLICATIVE NOISE

We prove the pathwise Strichartz and local smoothing estimates for a general class of stochastic dispersive equations with linear multiplicative noise, including especially the stochastic Schrodinger and Airy equations. The upper bounds and high integrability of random constants in these estimates are also obtained. These estimates have several applications to stochastic nonlinear problems, including the pathwise well-posedness and P-integrability of solutions. Furthermore, the large deviation principle for the small noise asymptotics is derived for both the stochastic inhomoge-neous dispersive equations and stochastic nonlinear Schrodinger equations with variable coefficients. The crucial point of proof is the invariance of the principal symbol under the Doss-Sussman type transformation. The lower order perturbations are controlled by the pseudodifferential calculus and semimartingale properties of stochastic homogeneous solutions.