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In this paper, we provide suitable adaptations of the "weak version ofBernstein method" introduced by the first author in 1991, in order to obtainLipschitz regularity results and Lipschitz estimates for nonlinearintegro-differential elliptic and parabolic equations set in the whole space.Our interest is to obtain such Lipschitz results to possibly degenerateequations, or to equations which are indeed "uniformly el-liptic" (maybe in thenonlocal sense) but which do not satisfy the usual "growth condition" on thegradient term allowing to use (for example) the Ishii-Lions' method. We treatthe case of a model equation with a superlinear coercivity on the gradient termwhich has a leading role in the equation. This regularity result together withcomparison principle provided for the problem allow to obtain the ergodic largetime behavior of the evolution problem in the periodic setting.

Lipschitz regularity for integro-differential equations with coercive hamiltonians and application to large time behavior