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We derive an approximate Gaussian solution of the Lindblad equation in thesemiclassical limit, given a general Hamiltonian and linear coupling with theenvironment. The theory is carried out in the chord representation anddescribes the evolved quantum characteristic function, which gives directaccess to the Wigner function and the position representation of the densityoperator by Fourier transforms. The propagation is based on a system ofnon-linear equations taking place in a double phase space, which coincides withHeller's theory of unitary evolution of Gaussian wave packets when theLindbladian part is zero.

Markovian evolution of Gaussian states in the semiclassical limit