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In this work we establish a Freidlin-Wentzell type large deviation principlefor stochastic nonlinear Schrodinger equation with either focusing ordefocusing nonlinearity driven by nonlinear multiplicative Levy noise in theMarcus canonical form. One essential ingredient of the proof is a variant ofthe well-known large deviation criteria of Budhiraja, Chen, Dupuis, andMaroulas for functions of Poisson random measures, which is proposed by thethird named author of this paper and his collaborators (Potential Anal. 2022).Obtaining the large deviation principle result is challenging in the currentsetting due to the presence of the nonlinear term, the lack of regularizationeffect of the Schrodinger operator and the absence of compactness ofembeddings. To overcome these difficulties, we employ a regularizationprocedure based on Yosida approximations to handle the lack of regularizationeffect of the Schrodinger operator. Additionally, to tackle the absence ofcompact embeddings, we adopt several techniques including time discretization,cut-off arguments and relative entropy estimates of sequences of probabilitymeasures. This also distinguishes the current work from most of previousstudies on large deviation principle of stochastic partial differentialequations with Levy noise, which heavily rely on the compactness conditions.

Large deviation principles for stochastic nonlinear Schrodinger equations driven by Levy noise