  In historical mathematics and physics, the Kardar-Parisi-Zhang equation or a
quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi
equation in growing interface and universality classes, is also known by the
different name as the quasilinear Riccati type equation. The existence of
solutions to this type of equation under some assumptions and requirements,
still remains an interesting open problem at the moment. In our previous
studies \cite{MP2018, MPT2019}, we obtained the global bounds and gradient
estimates for quasilinear elliptic equations with measure data. There have been
many applications are discussed related to these works, and main goal of this
paper is to obtain the existence of a renormalized solution to the quasilinear
stationary solution to the degenerate diffusive Hamilton-Jacobi equation with
the finite measure data in Lorentz-Morrey spaces.
