In this work, we develop a kinetic theory of stimulated Brillouin scattering (SBS) based on a generalized Bhatnagar-Gross-Krook (BGK) model incorporating a velocity-dependent collision frequency. Starting from the Boltzmann equation, we derive an analytic expression for the SBS gain coefficient that explicitly captures the coupling between velocity-dependent relaxation and acoustic response. We show that this modification leads to a systematic reshaping of the gain spectrum, including broadening, peak enhancement, and pressure-dependent frequency shifts that are not captured by conventional BGK theory. Our results demonstrate improved agreement with experimentally measured gain coefficients, identifying velocity-resolved relaxation as a key physical mechanism underlying SBS in rarefied and intermediate collisional regimes. This framework provides a physically grounded extension of kinetic SBS theory and introduces a parameter for controlling and optimizing Brillouin gain in high-power laser systems.

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