  Non-equilibrium noise is characterized as noise realizations where external
agitations disrupt the harmonic equilibrium of Brownian motion. Excitations in
a particle's random walk into a so-called L\'evy flight changes the
distribution of the noise from Gaussian to the fat-tailed L\'evy distribution.
Generalization between Gaussian and L\'evy distributions is the
$\alpha$-stability distribution, where $1<\alpha\leq2$. In this study, the
$\alpha$-stability distributed noise is subjugated into the Langevin and
fractional Fokker--Planck equations that correspond to a V-shaped linear
potential well $V(x)=F|x|$. From these equations, an Euler scheme for
computational simulation via iterations is presented, and a probability density
function that is normalizable under any $\alpha\in(1,2]$ is obtained. This
study is focused more on the theoretical framework of non-equilibrium noise in
V-shaped linear well profiles, which is intended to be applied to systems known
to exhibit self-organized criticality.
