Optimal conditions for first passage of jump processes with resetting

TL;DR

Addresses how to minimize the mean first passage time to cross a barrier for a discrete-time jump process under stochastic resetting with restart probability $r$. The authors derive a renewal-equation-based MFPT formula that depends on two-step reset-free statistics, enabling a simple criterion for the existence of an optimal $0<r^*<1$. They verify the criterion on two classes of jumps—skewed Laplace distributions and symmetric Lévy stable jumps—producing phase diagrams that separate regions with nontrivial resetting, resetting after every jump, or no resetting. The results show that nontrivial resetting can improve MFPT in wide parameter ranges, including all $0<\alpha<2$ Lévy indices beyond a threshold distance, and provide actionable guidance for designing optimal search protocols. The work also outlines extensions to more general reset mechanisms.

Abstract

We investigate the first passage time beyond a barrier located at $b\geq0$ of a random walk with independent and identically distributed jumps, starting from $x_0=0$. The walk is subject to stochastic resetting, meaning that after each step the evolution is restarted with fixed probability $r$. We consider a resetting protocol that is an intermediate situation between a random walk ($r=0$) and an uncorrelated sequence of jumps all starting from the origin ($r=1$), and derive a general condition for determining when restarting the process with $0<r<1$ is more efficient than restarting after each jump. If the mean first passage time of the process in absence of resetting is larger than that of the sequence of jumps, this condition is sufficient to establish the existence of an optimal $0<r^*<1$ that represents the best strategy, outperforming both $r=0$ and $r=1$. Our findings are discussed by considering two important examples of jump processes, for which we draw the phase diagram illustrating the regions of the parameter space where resetting with some $0<r^*<1$ is optimal.

Figures (2)

• Figure 1: Phase diagram for random walks with jumps drawn from the skewed Laplace distribution, with $\lambda(\eta)$ given by Eq. \ref{['eq:PDF_exp_bias']}. We set $b/a_+=X$ and $a_-/a_+=Y$. Region I is the domain where there exists a nontrivial $0<r^*<1$ that minimizes the $\langle \tau_{b}(r)\rangle$. In region II $\langle \tau_{b}(r)\rangle$ is a monotonic decreasing function of $r$, with global minimum at $r^*=1$. In region III instead we observe the opposite behaviour, and $\langle \tau_{b}(r)\rangle$ is a monotonic increasing function of $r$, attaining its global minimum at $r^*=0$. Note that this region extends in the $Y$-direction only as long as $a_-/a_+<1$, i.e., when the jump distribution is biased toward the barrier. Region I is delimited by the critical curves $\overline{Y}_1(X)$ and $\overline{Y}_2(X)$ given by Eq. \ref{['eq:Y1']} and \ref{['eq:Y3']} respectively, and contains the dashed curve representing $Y(X)$ given by Eq. \ref{['eq:Y2']}, which splits I into two subregions, according to whether $\langle \tau_{b}(0)\rangle>\langle \tau_{b}(1)\rangle$ (between $\overline{Y}_1(X)$ and $Y(X)$) or vice versa (between $Y(X)$ and $\overline{Y}_2(X)$). Panels (a), (b) and (c) show several examples of MFPT curves. The data are the results of numerical simulation and are compared to the exact curves given by Eq. \ref{['eq:BiasExp_MFPT_exact']}, showing excellent agreement.
• Figure 2: Phase diagram for Lévy flights with jumps drawn from Lévy stable laws, with $\hat{\lambda}(k)=e^{-|\gamma k|^\alpha}$. Region I is the domain where Eq. \ref{['eq:Cond']} is verified, so $\langle \tau_{b}(r)\rangle$ is guaranteed to have its global minimum for some $0<r^*<1$. In region II instead the existence of a nontrivial resetting probability is not guaranteed, and indeed in this domain our numerical simulations show a monotonic behaviour of $\langle \tau_{b}(r)\rangle$, suggesting that the minimum is always attained at $r^*=1$. Plots (a) and (b) show examples (with $\alpha=1.7$ and $\alpha=0.9$ respectively) of MFPT curves. The phase diagram does not show values $\alpha<0.7$ due to numerical difficulties in computing the critical point where region I begins, see text.