We analytically investigate the noise-driven response of a mechanical Duffing oscillator. Using a perturbative expansion within the Martin–Siggia–Rose–Janssen–De Dominicis path-integral formalism, we show that the fluctuation-dissipation theorem (FDT) extends to the nonlinear regime. Within renormalization theory and Feynman diagrammatic techniques, we derive the self-energy of the Duffing oscillator and the corresponding renormalized noise intensity, demonstrating the manifestation of the FDT in direct analogy with the linear oscillator. Furthermore, we illustrate how the Duffing nonlinearity enables the injection of energy by noise, allowing the excitation of higher harmonics.

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• Classical statistical mechanics
• Fluctuation-dissipation theorem
• Nonlinear beam dynamics
• Markovian processes
• Path-integral methods
• Stochastic differential equations