<p>In this paper, we focus on a Hirota-type equation, which is constructed by introducing Brownian motion and Gaussian white noise terms into the Hirota equation. First, the bilinear form of the equation is derived using the Hirota method. Based on the perturbation method, the first-order, second-order, and third-order soliton solutions are obtained respectively. By adjusting the parameters, various forms of analytical solutions are obtained, and the influence of parameter variations on the solutions is analyzed with the help of soliton plots. Under the effect of perturbations, a single soliton can exhibit interaction behavior similar to that of two solitons, while two solitons display dynamical characteristics resembling those of three solitons, and so on. Furthermore, different types of perturbations induce distinct interaction patterns of solitons. Meanwhile, the selection of different parameters can trigger breather phenomena or coupling interactions between breathers and solitons. Additionally, the constraint conditions for the Lax pair and the potential function are given, and the elliptic-rogue waves are constructed based on the Darboux transformation. These two types of periodic rogue waves correspond to the cn background and dn background of Jacobi elliptic functions, respectively, illustrating the excitation characteristics of rogue waves under different periodic modulation conditions. Finally, we employ the linear stability analysis to investigate the stability of the soliton solution.</p>