gains a gap of exactly $\Gamma$, with the corresponding stationary state given by the limit of Eq. , with probabilities $$\label{AsymHopSS}
P_\text{ss}^\Gamma(x) = \frac{\Gamma}{\gamma+\Gamma}{\left(\frac{\gamma}{\gamma+\Gamma}\right)}^{x-1},$$ where normalization can be checked via the geometric series. In this way, we see that resets localise the particle near the reset state, with a localisation length of $$\epsilon = \frac{1}{\ln\left(\frac{\gamma+\Gamma}{\gamma}\right)}.$$

Brownian motion {#BMReset}
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We now show how to apply our results to continuous-state models by seeing them as the limit of a sequence of finite-dimensional models which have a discrete spectrum and well-defined eigenvectors. We consider for this purpose the reset Brownian motion in one dimension, first studied in [@evans2011] via a modified Fokker-Planck equation.

The master, or Fokker-Planck operator in this case, without reset is the Laplacian $${\mathcal{L}}p(x)=D\frac{d^2 p(x)}{d x^2}$$ on the real line, with $D$ as the diffusion constant and $p(x)$ the probability density. The spectrum of this operator is trivial, but it has no normalised stationary state, nor is it possible to define the orthogonality relation between left and right eigenfunctions. To address this problem, we restrict the system, as commonly done in physics, to the finite interval $[-L/2,L/2)$ with periodic boundary conditions. The spectrum of this restricted model is simply 