Dynamical properties of the Bose-Einstein condensate in double-well potential subject to Gaussian white noise are investigated by numerically solving the time-dependent Gross-Pitaevskii equation. The Gaussian white noise is used to describe influence of the random environmental disturbance on the double-well condensate. Dynamical evolutions from three different initial states, the Josephson oscillation state, the running phase and $pi$-mode macroscopic quantum self-trapping states are considered. It is shown that the system is rather robust with respect to the weak noise whose strength is small and change rate is high. If the evolution time is sufficiently long, the weak noise will finally drive the system to evolve from high energy states to low energy states, but in a manner rather different from the energy-dissipation effect. In presence of strong noise with either large strength or slow change rate, the double-well condensate may exhibit very irregular dynamical behaviors.
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