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We analyze the non-Markovian dynamics of a quantum system subject tospontaneous collapse in space. After having proved, under suitable conditions,the separation of the center-of-mass and relative motions, we focus ouranalysis on the time evolution of the center of mass of an isolated system(free particle case). We compute the explicit expression of the Green'sfunction, for a generic Gaussian noise, and analyze in detail the case of anexponential correlation function. We study the time evolution of averagequantities, such as the mean position, momentum and energy. We eventuallyspecialize to the case of Gaussian wave functions, and prove that all basicfacts about collapse models, which are known to be true in the white noisecase, hold also in the more general case of non-Markovian dynamics.

Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties