To $\mathcal{P}\mathcal{T}$ or not to $\mathcal{P}\mathcal{T}$: Noise-induced escape and nonlinear-damping stabilization in a parity-time dimer

Parity-time ($\mathcal{P}\mathcal{T}$) symmetric systems exhibit long-lived excitations by balancing gain and loss in coupled resonators, driving extensive theoretical interest and diverse experimental realizations. Realistic physical implementations, however, inevitably introduce nonlinearities and noise. This mandates a rigorous reevaluation of their global long-time dynamics. In this work, we show that Hamiltonian Duffing nonlinearity restricts the $\mathcal{P}\mathcal{T}$-unbroken phase to a finite, nonattracting region of phase space. Consequently, unavoidable fluctuations drive first-passage escape into runaway trajectories. This renders the linearly $\mathcal{P}\mathcal{T}$-unbroken phase a purely transient phenomenon. We then recover global stochastic stability by introducing two-photon loss on the gain oscillator. This nonlinear damping explicitly breaks exact $\mathcal{P}\mathcal{T}$ symmetry while supplying genuine phase-space attraction, generating a bistable regime where a low-amplitude orbit mimicking the original linear state coexists with a high-amplitude limit cycle. Thus, we establish a revised origin for stability in non-Hermitian experiments: the observed long-time stochastic stability is governed by inherent restoring dissipation rather than the spectral $\mathcal{P}\mathcal{T}$ symmetry itself.