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In this paper, we establish large deviation principle for the strong solutionof a doubly nonlinear PDE driven by small multiplicative Brownian noise.Motononicity arguments and the weak convergence approach have been exploited inthe proof. Moreover, by using certain a-priori estimates and sequentiallyweakly Feller property of the associated Markov semigroup, we show existence ofinvariant probability measure for the strong solution of the underlyingproblem.

Stochastic doubly nonlinear PDE: Large Deviation Principles and existence of Invariant measure