Directed random walk with random restarts: The Sisyphus random walk

In this paper we consider a particular version of the random walk with restarts: random reset events which suddenly bring the system to the starting value. We analyze its relevant statistical properties, like the transition probability, and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks, and other extreme statistics are also derived; we consider counting problems naturally associated with the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.

Figure 1

Typical realization of the process $X_t$ for $q_{\ell }=4/5$, independent of $\ell$.

Figure 2

Distribution of the process $X_t$ at time $t=7$ for $q=4/5$. Note that the probability is decreasing, but the most likely position is found at the ending point $\ell =t$.

Figure 3

The conditional mean position of the process, $\mathbb{E}\left[\left.X_t\right|X_{t_0}={\ell }_0\right]$, is plotted as a function of $\tau =t-t_0$ for ${\ell }_0=2$ and $q=4/5$.

Figure 4

First-passage time probability of the processes $X_t$ for $\ell =10$ and $q=4/5$.

Figure 5

Realization of the process $M_t$ for the sample path $X_t$ shown in Fig. 1.

Figure 6

High-water mark probability of the processes $X_t$ for $t=30$ and (a) $q=4/5$, (b) $q=9/10$.

Figure 7

A feasible realization of the process $X_t$ with shrinking probabilities: $q_{\ell }=q_0/(\ell +1)$ and $q_0=4/5$.