Probabilistic representations of deterministic partial differential equations often provide important insight into the underlying physical processes, e.g. Brownian motion as a model for the motion of indivial particles in a solute whose concentration satisfies the advection-diffusion equation. Recently, a family of probabilistic representations for the solution to the Navier-Stokes equation (NSE) has been discovered. This family is parametrized by a collection of functions dubbed "majorizing kernels." For any such majorizing kernel h, the Fourier transform of the solution to NSE is written as the expected value of a random multiplicative process defined on a tree in Fourier space, where the contribution of each frequency to the solution is determined by h. Even though this structure irresistibly suggests a connection with the energy cascade model of turbulent flows, no firm grounds for a physical interpretation of the probabilistic representation exist. In particular, the construction of majorizing kernels for which the spectrum of the solution exhibits the scaling properties predicted by the turbulent cascade model, would be very desirable. In this talk I will show some elementary ongoing work on the formulation of this problem for Burgers equation: a simpler nonlinear model in fluid dynamics exhibiting analogous structure as NSE in its probabilistic representation and spectral properties.