  The well known phenomenon of exponential contraction for solutions to the
viscous Hamilton-Jacobi equation in the space-periodic setting is based on the
Markov mechanism. However, the corresponding Lyapunov exponent $\lambda(\nu)$
characterizing the exponential rate of contraction depends on the viscosity
$\nu$. The Markov mechanism provides only a lower bound for $\lambda(\nu)$
which vanishes in the limit $\nu \to 0$. At the same time, in the inviscid case
$\nu=0$ one also has exponential contraction based on a completely different
dynamical mechanism. This mechanism is based on hyperbolicity of
action-minimizing orbits for the related Lagrangian variational problem.
  In this paper we consider the discrete time case (kicked forcing), and
establish a uniform lower bound for $\lambda(\nu)$ which is valid for all
$\nu\geq 0$. The proof is based on a nontrivial interplay between the dynamical
and Markov mechanisms for exponential contraction. We combine PDE methods with
the ideas from the Weak KAM theory.
