Stochastic PDEs Driven by Nonlinear Noise and Backward Doubly SDEs

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We study a “new kind” of backward doubly stochastic differential equations, where the nonlinear noise term is given by Ito–Kunita's stochastic integral. This allows us to give a probabilistic interpretation of classical and Sobolev's solutions of semilinear parabolic stochastic partial differential equations driven by a nonlinear space-time noise.