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We prove some Liouville properties for sub- and supersolutions of fullynonlinear degenerate elliptic equations in the whole space. Our assumptionsallow the coefficients of the first order terms to be large at infinity,provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. Wegive two applications. The first is a stabilization property for large times ofsolutions to fully nonlinear parabolic equations. The second is the solvabilityof an ergodic Hamilton-Jacobi-Bellman equation that identifies a uniquecritical value of the operator.

Liouville properties and critical value of fully nonlinear elliptic operators