Large deviations for Markov processes with resetting

Markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches, foraging, and population dynamics. Here we study the large deviations of time-additive functions or observables of Markov processes with resetting. By deriving a renewal formula linking generating functions with and without resetting, we are able to obtain the rate function of such observables, characterizing the likelihood of their fluctuations in the long-time limit. We consider as an illustration the large deviations of the area of the Ornstein-Uhlenbeck process with resetting. Other applications involving diffusions, random walks, and jump processes with resetting or catastrophes are discussed.

Figure 1

Dominant pole of ${\tilde{G}}_r(x,k,s)$ for increasing truncation orders: $m=0$ (blue), $m=2$ (purple), and $m=4$ (green). The black curve shows the convex ${\lambda }_r\left(k\right)$ obtained for $m\ge 6$, the dashed black curve the nonreset ${\lambda }_0\left(k\right)$, and the dashed gray curve the tail approximation of ${\lambda }_r\left(k\right)$ shown in Eq. (35). The parameters are $x_r=1,r=2,\gamma =1$, and $\sigma =1$.

Figure 2

Black curves represent $I_r\left(a\right)$ for $x_r=0,1,2$ (from left to right). The first two curves were obtained for $m=10$, while the last for $x_r=2$ was obtained for $m=20$. The dashed black curve shows the nonreset rate function $I_0\left(a\right)$ and the dashed gray curve the tail approximation of $I_r\left(a\right)$ shown in Eq. (36). The parameters are $r=2,\gamma =1$, and $\sigma =1$.

Figure 3

Proportionality coefficient $c\left(r\right)$ between the minimum $a^{*}$ of $I_r\left(a\right)$ and the resetting position $x_r$. The parameters are $\gamma =1$ and $\sigma =1$.