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The classical Feynman-Kac identity represents solutions of linear partialdifferential equations in terms of stochastic differential euqations. Thisrepresentation has been generalized to nonlinear partial differential equationson the one hand via backward stochastic differential equations and on the otherhand via stochastic fixed-point equations. In this article we generalize therepresentation via stochastic fixed-point equations to allow the nonlinearityin the semilinear partial differential equation to depend also on the gradientof the solution.

On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear  parabolic partial differential equations with gradient-dependent  nonlinearities