Martingale Solutions Of Fractional Stochastic Evolution Equations On A Bounded Interval

In this paper, we study the existence of martingale solutions of fractional s-tochastic evolution equations on a bounded interval, and primarily consider the fractional s-tochastic nonlinear Schrodinger equation and the fractional stochastic reaction-diffusion equa-tion.For the fractional stochastic nonlinear Schrodinger equation on a bounded interval, we introduce a suitable function to construct the weighted fractional Sobolev space. Then, applying some fractional operator skills, it overcomes the diffculties caused by the fractional Laplacian operator on a bounded interval. Using the tightness instead of the common compactness, it solves the trouble of losing the common compactness of the system caused by the white noise.Furthermore, combining Prokhorov theorem with Skorokhod embedding theorem, it deals with the convergence problem. On the basis of a series of inequalities skills, we discuss in detail that when the coefficient of nonlinear term A is positive and negative respectively, the corresponding index of nonlinear term σ has different ranges. We finally establish the existence of martingale solution for the fractional stochastic nonlinear Schrodinger equation on a bounded interval.Using the similar method, we also obtain the existence of martingale solution of the fractional stochastic reaction-diffusion equation on a bounded interval.