Amplification induced by white noise

We investigate the amplification of the field induced by white noise. In the present study, we study a stochastic equation which has two parameters, the energy $\omega(\vec{k})$ of a free particle and the coupling strength $D$ between the field and white noise, where the quantity $\vec{k}$ represents the momentum of a free particle. This equation is reduced to the equation with one parameter $\alpha(\vec{k})$ which is defined as $\alpha(\vec{k}) = D (\omega(\vec{k}))^{-3/2}$. We obtain the expression of the exponent statistically averaged over the unit time and derive an approximate expression of it. In addition, the exponent is obtained numerically by solving the stochastic equation. We find that the amplification increases with $\alpha(\vec{k})$. This indicates that white noise can amplify the fields for soft modes if the mass $m$ of the field is sufficiently light and if the strength of the coupling between the field and white noise is sufficiently strong, when the energy $\omega(\vec{k})$ is equal to $\sqrt{m^{2} + \vec{k}^{2}}$. We show that the $\alpha(\vec{k})$ dependence of the exponent statistically averaged is qualitatively similar to that of the exponent obtained by solving the stochastic equation numerically, and that these two exponents for the small value of $\alpha(\vec{k})$ are quantitatively similar.