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On the long time behavior of stochastic Schroedinger evolutions

We discuss the time evolution of the wave function which is solution of a stochastic Schroedinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. Observing that there exist three time regimes, namely the collapse regime, after which the wave function is localized in space; the classical regime, during which the collapsed wave function moves along a classical path and the diffusive regime, in which diffusion overlaps significantly the deterministic motion, we study the long time behavior of the wave function. We assert that the general solution converges a.s. to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this.