  Starting from the stochastic Zakharov-Kuznetsov equation, a multidimensional
KdV type equation on a hypercubic lattice, we provide a derivation of the
3-wave kinetic equation. We show that the two point correlation function can be
asymptotically expressed as the solution of the 3-wave kinetic equation at the
kinetic limit under very general assumptions: the initial condition is out of
equilibrium, the dimension is $d\ge 2$, the smallness of the nonlinearity
$\lambda$ is allowed to be independent of the size of the lattice, the weak
noise is chosen not to compete with the weak nonlinearity and not to inject
energy into the equation. Unlike the cubic nonlinear Schr\"odinger equation,
for which such a general result is commonly expected without the noise, the
kinetic description of the deterministic lattice ZK equation is unlikely to
happen. One of the key reasons is that the dispersion relation of the lattice
ZK equation leads to a singular manifold, on which not only $3$-wave
interactions but also all $m$-wave interactions ($m\ge3$) are allowed to
happen. This phenomenon has been first observed by Lukkarinen
\cite{lukkarinen2007asymptotics} as a counterexample for which one of the main
tools to derive kinetic equations from wave equations (the suppression of
crossings) fails to hold true. To the best of our knowledge, the work provides
the first rigorous derivation of nonlinear 3-wave kinetic equations. Also this
is the first derivation for wave kinetic equations in the lattice setting and
out-of-equilibrium.
