关于Hilbert空间扩散的长时间行为
On the long time behavior of Hilbert space diffusion
摘要：
Hilbert空间中的随机微分方程作为随机非线性修改的Schroedinger方程近年来引起了人们的极大关注，特别令人感兴趣的是其解的长时间行为。本文讨论了描述自由量子粒子在空间中自发坍缩的时间演化的随机微分方程解的长时间行为。我们解释了为什么这个问题是微妙的，并报告最近严格的结果，它断言，任何初始状态几乎肯定收敛到一个高斯状态具有固定的传播位置和动量。
英文摘要：
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.
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