  Stability of the traveling wave solution to a general class of
one-dimensional nonlocal evolution equations is studied in $L^2$-spaces,
thereby providing an alternative approach to the usual spectral analysis with
respect to the supremum norm. We prove that the linearization around the
traveling wave solution satisfies a Lyapunov-type stability condition in a
weighted space $L^2(\rho)$ for a naturally associated density $\rho$. The
result can be applied to obtain stability of the traveling wave solution under
stochastic perturbations of additive or multiplicative type. For small wave
speeds, we also prove an alternative Lyapunov-type stability condition in
$L^2(\mathfrak{m})$, where $\mathfrak{m}$ is the symmetrizing density for the
traveling wave operator, which allows to derive a long-term stochastic
stability result.
