  We study a system of semilinear hyperbolic equations passively advected by
smooth white noise in time random velocity fields. Such a system arises in
modeling non-premixed isothermal turbulent flames under single-step kinetics of
fuel and oxidizer. We derive closed equations for one-point and multi-point
probability distribution functions (PDFs) and closed form analytical formulas
for the one point PDF function, as well as the two-point PDF function under
homogeneity and isotropy. Exact solution formulas allows us to analyze the
ensemble averaged fuel/oxidizer concentrations and the motion of their level
curves. We recover the empirical formulas of combustion in the thin reaction
zone limit and show that these approximate formulas can either underestimate or
overestimate average concentrations when reaction zone is not tending to zero.
We show that the averaged reaction rate slows down locally in space due to
random advection induced diffusion; and that the level curves of ensemble
averaged concentration undergo diffusion about mean locations.
