Approximation of mild solutions of a semilinear fractional differential equation with random noise

We study for the first time the Cauchy problem for semilinear fractional elliptic equations with random data. This paper is concerned with the Gaussian white noise model for initial Cauchy data. We establish the ill-posedness of the problem. Then we propose the Fourier truncation method for stabilizing the ill-posed problem. Some convergence rates between the exact solution and the regularized solution is established in L 2 L^2 and H q H^q norms.