Dynamics of Partial Differential Equations

Volume 6 (2009)

Number 2

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

Pages: 167 – 183

DOI: http://dx.doi.org/10.4310/DPDE.2009.v6.n2.a2

Author

Vahagn Nersesyan (Laboratoire de Mathématiques, Université de Paris-Sud, Orsay, France)

Abstract

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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