Exponential mixing for finite-dimensional approximations of the Schrödinger equation with multiplicative noise

Details Exponential mixing for finite-dimensional approximations of the Schrödinger equation with multiplicative noise Exponential mixing for finite-dimensional approximations of the Schrödinger equation with multiplicative noise

2007-10-19

arxiv.org/abs/0710.3693v1

Physics

Mathematical Physics

Scientific paper

We study the ergodicity of finite-dimensional approximations of the Schr\"odinger equation. The system is driven by a multiplicative scalar noise. Under general assumptions over the distribution of the noise, we show that the system has a unique stationary measure $\mu$ on the unit sphere $S$ in $\C^n$, and $\mu$ is absolutely continuous with respect to the Riemannian volume on $S$. Moreover, for any initial condition in $S$, the solution converges exponentially fast to the measure $\mu$ in the variational norm.

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