Fixed points of smoothing transformation in random environment

At each time n\in \mathbb{N},\mathrm{l}\mathrm{e}\mathrm{t}?\text{?}{\overline{Y}}^{(n)}(\xi )=(y_1^{(n)}(\xi ),y_2^{(n)}(\xi ),?) be a random sequence of non-negative numbers that are ultimately zero in a random environment\xi \text{=}（{\xi }_n{）}_{n\in \mathbb{N}}. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.\xi ,Z(\xi )\stackrel{\text{d}}{=}{\sum }_{i\in ?+}{y}_i^{(\mathrm{0})}(\xi )Z_i^{(\mathrm{1})}(\xi )，where ｛Z_i^{(\mathrm{1})}:i\in {?}_{+}｝ are random variables in random environment which satisfy that for any environment\xi; under P_{\xi }; ｛Z_i^{(\mathrm{1})}:i\in {?}_{+}｝are independent of each other and {\bar{Y}}^{(\mathrm{0})}(\xi ), and have the same conditional distribution P_{\xi }(Z_i^{(1)}(\xi )\in ?)=P_{T\xi }(Z(T\xi )\in ?) where T is the shift operator. This extends the classical results of J. D. Biggins J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.