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In this paper, we establish large deviation principle for the strong solutionof evolutionary p-Laplace equation driven by small multiplicative Browniannoise, where the weak convergence approach plays a key role. Moreover, by usingGirsanov transformation along with $L^1$-contraction approach, we show thequadratic transportation cost inequality for the strong solution of theunderlying problem.

Stochastic evolutionary p-Laplace equation: Large Deviation Principles and Transportation Cost Inequality