We present computation of thermal conductivity in one dimensional dispersive luttinger liquids

in the framework of kinetic theory. The decay rates for the bosonic or fermionic excitations of the

system are predicted from a diagonal approximation of the full collision integral operator. Analytical

and numerical support for this approximation is provided. Using these decay rates, we compute the

thermal conductivity in different regimes controlled by the frequency and temperature. At the lowest

frequencies, we observe connections to KPZ exponent that arise in dynamical RG of

one dimensional hydrodynamics.