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We present in this Letter a detailed analysis of mechanisms by which the phase space Brownian motion of an ensemble of massive particles, in the diffusion regime, is governed by the Schrödinger equation. It is explicitly shown how the pressure of the diffusing ensemble is linked to the quantum potential, known to appear in the Hamiltonian-Jacobi-Madelung formulation of the corresponding quantum dynamics. The quantum state vector (wave function) corresponds in this picture to the physically real diffusing medium governing the collective evolution of the particle ensemble.