Effects of white noise on parametric resonance in $\lambda \phi^{4}$ theory

We investigate the effects of white noise on parametric resonance in $\lambda \phi^{4}$ theory. The potential $V(\phi)$ in this study is ${1/2} m^{2} \phi^{2} + {1/3} g \phi^{3} + {1/4} \lambda \phi^{4}$. An Mathieu-like equation is derived and the derived equation is applied to a partially thermalized system. The magnitudes of the amplifications are extracted by solving the equations numerically for various values of parameters. It is found that the amplification is suppressed by white noise in almost all the cases. However, in some $g=0$ cases, the amplification with white noise is slightly stronger than that without white noise. In the $g=0$ cases, the fields are always amplified. The amplification is maximal at $k_{m} \neq 0$ in some $g=0$ cases. Contrarily, in the $g = {3 \sqrt{2 \lambda} m}/{2}$ cases, the fields for some finite modes are suppressed and the amplification is maximal at $k_{m} \sim 0$ when the amplification occurs. It is possible to distinguish by these differences whether the system is on the $g=0$ state or not.