Quantum Brownian Motion as a Classical Stochastic Process in Phase Space

We establish that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model, with at most quadratic external potential, can be mapped, at any temperature, onto a classical, non-Markovian stochastic process in phase space. Starting from a correlated thermal equilibrium state between the particle and bath, we demonstrate that this correspondence is exact for quadratic potentials under arbitrary quantum state preparations of the particle itself. Our approach allows to consider arbitrary initial quantum states - including highly non-classical superpositions - which are incorporated via their Wigner functions, which serve as statistical weights for trajectory ensembles. Furthermore, the formalism naturally accommodates external manipulations and measurements modeled by preparation functions acting at arbitrary times, enabling the simulation of complex driven-dissipative quantum protocols. For more general, smooth potentials, we identify a natural small parameter: the density matrix becomes strongly quasidiagonal in the coordinate representation, with its off-diagonal width shrinking as the bath's spectral cutoff increases, suggesting a controlled parameter for a possible approximation.