Dynamics of Partial Differential Equations
Volume 6 (2009)
Exponential mixing for finite-dimensional approximations of the Schrödinger equation with multiplicative noise
Pages: 167 – 183
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.
exponential mixing, ergodicity, Schrödinger equation, multiplicative noise
2010 Mathematics Subject Classification
34-xx, 35-xx, 37-xx