On the wave turbulence theory for a stochastic KdV type equation -- Generalization for the inhomogeneous kinetic limit

Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension $d\ge 2$, 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. In the present work, we build on the framework of [ST21], following the formal derivation of Spohn [Spo06] and inspired by previous work [HO21] of the first author and Olla, so that the inhomogeneous 3-wave kinetic equation can also be obtained at the kinetic limit under analogous assumptions. Similar to the homogeneous case -- and unlike the cubic nonlinear Schr\"odinger equation -- the inhomogeneous kinetic description of the deterministic lattice ZK equation is unlikely to happen due to the vanishing of the dispersion relation on a certain singular manifold on which not only $3$-wave interactions but also all $n$-wave interactions ($n\ge3$) are allowed to happen, a phenomenon first observed by Lukkarinen [Luk07]. To the best of our knowledge, our work provides the first rigorous derivation of a nonlinear inhomogeneous wave kinetic equation in the kinetic limit.