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

We study the ergodicity of finite-dimensional approximations of the Schrödinger 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 μ on the unit sphere S in Cⁿ, and μ 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 μ in the variational norm.

Keywords

exponential mixing, ergodicity, Schrödinger equation, multiplicative noise

2010 Mathematics Subject Classification

34-xx, 35-xx, 37-xx

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Published 1 January 2009