parameters of [Eq. (\[eq:lgv2\])]{}, $F$, $B$ and $\tau$, which can be obtained, by fitting the first three cumulants of $P_{yx}$, $\kappa_i(P_{yx}), i=1,2,3$, in a nonequilibrium steady-state $\gamma\ne0$, as follows.

Formulas for the steady-state cumulants of $\alpha(t)$, which evolves according to [Eq. (\[eq:lgv2\])]{}, can be derived, by using the method of Ref. [@PRE2016III Appendix A]. For this we need to consider a random variable $R(t)$, given by the time integral of the force terms in the Langevin dynamics: $${\label{eq:Rt}} R(t) = \int_0^{t} d{t^\prime} [F + A \omega({t^\prime}) + B \epsilon({t^\prime})].$$ The steady-state cumulants of $\alpha(t)$ can be then expressed as $$\begin{aligned}
{\label{eq:kappar}}
    \kappa_1(\alpha) &=& \frac{\kappa_1(R)}{b^2 t};\quad
    \kappa_2(\alpha) = \frac{\kappa_2(R)}{2 a b^2 t};\nonumber\\
    \kappa_3(\alpha) &=& \frac{2 \kappa_3(R)}{3 b^2 (2 a^2 + b^2) t}.\end{aligned}$$ cf. Ref. [@PRE2016III Appendix A].

Equation (\[eq:kappar\]) can be further expanded, once the form of athermal noise in [Eq. (\[eq:lgv2\])]{}, $B \epsilon(t/\tau)$, is specified. We discussed in [Sec. \[sec:intro\]]{}, that in this paper we consider separately two models, simple exponential noise ($\mathrm{ex}$) and exponential shot noise ($\mathrm{sh}$). From now on, to distinguish them we will use subscripts in the parameters of [Eq. (\[eq:lgv2\])]{}, respectively, $F_\mathrm{ex} + B_\mathrm{ex} \epsilon_\mathrm{ex}(t/\tau_\mathrm{ex})$ and $F_\mathrm{sh} + B_\mathrm{sh} \epsilon_\mathrm{sh}(t/\tau_\mathrm{sh})$. In the case of simple exponential 