and the reset rates magnitude relative to the corresponding eigenvalue $\lambda_i$. This agrees with intuition: since the eigenvalues are related to the lifetimes of the dynamical modes, if the average time between resets is larger than the lifetime of a mode, it will not make a significant contribution to the new steady state. We note that this equation also has the expected limits of $\ket{P^0_\text{ss}}=\ket{P_\text{ss}}$ and $\ket{P^\infty_\text{ss}}=\ket{C_0}$.

This result applies for processes with a finite number $D$ of states, but also to infinite-dimensional processes, provided that they possess a well-defined spectrum of eigenvalues with corresponding left and right eigenmodes. This latter fact will be illustrated in the next section with the example of Brownian motion.

We should also note that the result holds if the reset state $\ket{C_0}$ is replaced by a reset probability distribution $\ket{P_0}$ over configurations, giving the probability of reaching different states after a reset event. In this case, $\ket{C_0}$ in Eq.  is simply replaced by the “mixed” reset state $\ket{P_0}$. This follows since the reset state is never referred to above as anything more than as a vector in state space.

Finally we mention the modification resulting from a lack of ergodicity, i.e., when $\mathcal{L}\ket{r_i}=0$ and 