  The main goal of this paper is to obtain sufficient conditions that allow us
to rigorously derive local versions of the 4/5 and 4/3 laws of hydrodynamic
turbulence, by which we mean versions of these laws that hold in bounded
domains. This is done in the context of stationary martingale solutions of the
Navier-Stokes equations driven by an Ornstein-Uhlenbeck process. Specifically,
we show that under an assumption of \say{on average} precompactness in $L^3,$
the local structure functions are expressed up to first order in the length
scale as nonlinear fluxes, in the vanishing viscosity limit and within an
appropriate range of scales. If in addition one assumes local energy equality,
this is equivalent to expressing the structure functions in terms of the local
dissipation. Our precompactness assumption is also shown to produce stationary
martingale solutions of the Euler equations with the same type of forcing in
the vanishing viscosity limit.
