Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations

Details Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations

2006-02-16

arxiv.org/abs/math/0602350v1

The Annals of Probability 36, 3 (2008) 896-930

Mathematics

Numerical Analysis

Scientific paper

10.1214/07-AOP344

We consider weakly damped nonlinear Schr\"odinger equations perturbed by a noise of small amplitude. The small noise is either complex and of additive type or real and of multiplicative type. It is white in time and colored in space. Zero is an asymptotically stable equilibrium point of the deterministic equations. We study the exit from a neighborhood of zero, invariant by the flow of the deterministic equation, in $\xLtwo$ or in $\xHone$. Due to noise, large fluctuations off zero occur. Thus, on a sufficiently large time scale, exit from these domains of attraction occur. A formal characterization of the small noise asymptotic of both the first exit times and the exit points is given.

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