Characterization of stationary states in random walks with stochastic resetting

It is known that introducing a stochastic resetting in a random-walk process can lead to the emergence of a stationary state. Here we study this point from a general perspective through the derivation and analysis of mesoscopic (continuous-time random walk) equations for both jump and velocity models with stochastic resetting. In the case of jump models it is shown that stationary states emerge for any shape of the waiting-time and jump length distributions. The existence of such state entails the saturation of the mean square displacement to an universal value that depends on the second moment of the jump distribution and the resetting probability. The transient dynamics towards the stationary state depends on how the waiting time probability density function decays with time. If the moments of the jump distribution are finite then the tail of the stationary distributions is universally exponential, but for Lévy flights these tails decay as a power law whose exponent coincides with that from the jump distribution. For velocity models we observe that the stationary state emerges only if the distribution of flight durations has finite moments of lower order; otherwise, as occurs for Lévy walks, the stationary state does not exist, and the mean square displacement grows ballistically or superdiffusively, depending on the specific shape of the distribution of movement durations.

Figure 1

$P_s\left(x\right)$ with a $\mathrm{\Phi }\left(x\right)$ given by the Gaussian PDF for different values of the probability of resetting $r$. The results calculated from Eq. (10) have been drawn with symbols $\sigma =1$. Stochastic simulations directly performed for Eqs. (1) and (5) are shown with filled circles exhibiting an excellent agreement. The solid curves indicate the approximations to the tails and the central part of $P_s\left(x\right)$.

Figure 2

(a) $P_s\left(x\right)$ (symbols) for the Cauchy jump PDF for different values of the probability of resetting $r$. The symbols correspond to the solutions calculated from Eq. (10) taking $a=1$. Stochastic simulations directly performed for Eqs. (1) and (5) are shown with filled circles exhibiting an excellent agreement. (b) Log-log plot for $P_s\left(x\right)$ for the Cauchy jump PDF for different values of the probability of resetting $r$. The solid curves have been calculated from Eq. (10) taking $a=1$.

Figure 3

Plot of the MDS relaxing to the asymptotic value $(1-r)〈l^2〉/r$. In this case the jump PDF has finite moments but the waiting-time PDF is a power law, that is, $\langle x^2\rangle$ is given by Eq. (27). Here $\tau =〈l^2〉=1,r=0.1$.

Figure 4

Stationary distribution (symbols) for a PDF movement durations $\phi \left(t\right)=t{e}^{-t/\tau }/{\tau }^2$ for different values of $r$. Stochastic simulations directly performed for Eqs. (1) and (32) are shown with filled circles exhibiting an excellent agreement. The solid curves indicate the approximations to the tails and the central part of $P_s\left(x\right)$. $v_0=\tau =1$.