We develop a microscopic transport theory in a randomly driven fermionic model with and without a linear potential. The operator dynamics arise from the competition between noisy and static couplings, leading to diffusion regardless of ballistic transport or Stark localization in the clean limit. The universal diffusive behavior is attributed to a noise-induced bound state arising in the operator equations of motion at small momentum. By mapping the noise-averaged operator equation of motion to a one-dimensional non-Hermitian hopping model, we analytically solve for the diffusion constant, which scales nonmonotonically with noise strength, revealing regions of enhanced and suppressed diffusion from the interplay between on-site and bond dephasing noise, and a linear potential. For large on-site dephasing, the diffusion constant vanishes, indicating an emergent localization. On the other hand, the operator equation becomes the diffusion equation for strong bond dephasing and is unaffected by additional arbitrarily strong static terms that commute with the local charge, including density-density interactions. The bound state enters a continuum of scattering states at finite noise and vanishes. However, the bound state reemerges at an exceptional-like point in the spectrum after the bound-to-scattering state transition. We then characterize the fate of Stark localization in the presence of noise.

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• Diffusion
• Dissipative dynamics
• Noise
• Lindblad equation
• Tight-binding model