  We provide a semigroup approach to the viscous Hamilton-Jacobi equation. It
turns out that exponential Orlicz hearts are suitable spaces to handle the
(quadratic) non-linearity of the Hamiltonian. Based on an abstract extension
result for nonlinear semigroups on spaces of continuous functions, we represent
the solution of the viscous Hamilton-Jacobi equation as a strongly continuous
convex semigroup on an exponential Orlicz heart. As a result, the solution
depends continuously on the initial data. We further determine the symmetric
Lipschitz set which is invariant under the semigroup. This automatically yields
a priori estimates and regularity in Sobolev spaces. In particular, on the
domain restricted to the symmetric Lipschitz set, the generator can be
explicitly determined and linked with the viscous Hamilton-Jacobi equation.
