Abstract: (joint work with Loth Chabi) We study the Liouville type classification and symmetry properties,
in \(R^n\) and in a half-space with Dirichlet boundary conditions, for entire and ancient solutions of the diffusive Hamilton-Jacobi equation,
which arises in optimal stochastic control, in KPZ type models of surface growth and in studies of boundary gradient blow-up.

In particular we obtain optimal Liouville type theorems for ancient and entire solutions in \(R^n\) and we completely classify entire solutions in a half-space.
The proofs rely, among other things, on new and optimal, local estimates of Bernstein and Li-Yau type.
Such Liouville type results are very useful in the qualitative analysis of blow-up singularities for this equation.