  We study for the first time the Cauchy problem for semilinear fractional
elliptic equation. This paper is concerned with the Gaussian white noise model
for the initial Cauchy data. We establish the ill-posedness of the problem.
Then, under some assumption on the exact solution, 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 $L^2$
and $H^q$ norms.
