We study the asymptotics of front speeds of the reaction-diffusion
equations with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity
and zero mean stationary ergodic Gaussian shear advection on
the entire plane. By exploiting connections of KPP front speeds
with the almost sure Lyapunov exponents of the parabolic Anderson
problem, and with the homogenized Hamiltonians of Hamilton-Jacobi
equations, we show that front speeds enhancement is quadratic in
the small root mean square (rms) amplitudes of white in time zero
mean Gaussian shears, and it grows at the order of the large rms
amplitudes. However, front speeds diverge logarithmically if the
shears are time independent zero mean stationary ergodic Gaussian
fields.