<p>Nonlinear structural systems subjected to stochastic excitation exhibit complex probabilistic responses that cannot be predicted using deterministic formulations. Such behavior becomes particularly significant in graded nano-scale structures, where material inhomogeneity, geometric nonlinearity, and random environmental fluctuations interact to govern the global dynamics. The present work investigates the stochastic nonlinear response of porous functionally graded nanobeams under multiplicative excitation. A reduced-order nonlinear oscillator model is derived to capture the essential mechanics of the system, and the stationary response is characterized through the associated Fokker–Planck equation with state-dependent diffusion. Independent Monte-Carlo simulations are conducted to validate the predicted stationary probability density, while residual evaluation of the governing probability equation confirms solution consistency. The results show that the interaction between nonlinear stiffness and multiplicative stochastic forcing leads to bistable probability distributions and noise-induced transitions between stable states, indicating the occurrence of P-bifurcation. The proposed formulation provides a physically consistent and computationally efficient framework for predicting stochastic responses of graded nanostructures and contributes to improved understanding of probabilistic dynamics in nonlinear heterogeneous systems.</p>