On the long time behavior of Hilbert space diffusion

Stochastic differential equations in Hilbert space as random nonlinear modified Schroedinger equations have achieved great attention in recent years; of particular interest is the long time behavior of their solutions. In this note we discuss the long time behavior of the solutions of the stochastic differential equation describing the time evolution of a free quantum particle subject to spontaneous collapses in space. We explain why the problem is subtle and report on a recent rigorous result, which asserts that any initial state converges almost surely to a Gaussian state having a fixed spread both in position and momentum.Comment: 6 pages, EPL2-Te

On a general kinetic equation for many-particle systems with interaction, fragmentation and coagulation

We deduce the most general kinetic equation that describe the low density limit of general Feller processes for the systems of random number of particles with interaction, collisions, fragmentation and coagulation. This is done by studying the limiting as ε -> 0 evolution of Feller processes on ∪n∞ Xn with X = Rd or X = Zd described by the generators of the form ε-1 ∑K k=0 εkB(k), K ∈ N, where B(k) are the generators of k-arnary interaction, whose general structure is also described in the paper