  We establish a priori Lipschitz estimates for unbounded solutions of
second-order Hamilton-Jacobi equations in R^N in presence of an
Ornstein-Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii
\& Loreti (2006) in several directions. The first one is to consider more
general operators. We first replace the Laplacian by a general diffusion matrix
and then consider a nonlocal integro-differential operator of fractional
Laplacian type. The second kind of extension is to deal with more general
Hamiltonians which are merely sublinear. These results are obtained for both
degenerate and nondegenerate equations.

Analysis of PDEs