Minimization of the expected first-passage time of a Brownian motion with Poissonian resetting

TL;DR

The paper studies minimizing the expected first-passage time (FPT) and first-exit time (FET) of a Brownian motion with Poissonian resetting with respect to the resetting rate $r$, for the one-boundary and two-boundary geometries. It derives explicit expressions for the mean times $T(x,r)$, proves finiteness due to resetting, and identifies the optimal resetting rate $r_m(x)$ that minimizes $T(x,r)$; in the one-boundary case, $r_m(x)$ lies within $\big(\alpha,\beta\big)$ with $\alpha=\frac{1}{2 x_R^2}$ and $\beta=\frac{2}{x_R^2}$ and increases with the starting position $x$. In the two-boundary case on $(0,b)$ the optimal rate is governed by a closed-form in terms of hyperbolic sines and can exhibit a qualitative transition as $x_R$ crosses a critical value near $0.295$, including regimes where $r_m(x)=0$ is optimal for all $x$. The results indicate that resetting can expedite boundary-crossing tasks and motivate extensions to drifted Brownian motion, other diffusion processes, or numerical Monte Carlo methods when analytic expressions are unavailable.

Abstract

We address the problem of minimizing the expected first-passage time of a Brownian motion with Poissonian resetting, with respect to the resetting rate $r.$ We consider both the one-boundary and the two-boundary cases.

Figures (9)

• Figure 1: Graphs of $T(x,r),$ as a function of $r>0,$ for fixed $x_R=1$ and for $x=1$ (lower curve) and $x=200$ (higher curve); (on the horizontal axes $r).$
• Figure 2: Graphs of $T(x,r),$ as functions of $r>0,$ for fixed $x_R=1$ and for the values of $x$ contained in the first column of Table \ref{['tab2']} (on the horizontal axes $r);$ the point of mimimum $r_{m}(x)$ increases from $\alpha = \frac{1}{2x_R^2} = \frac{1}{2}$ (attained at $x = 0.0001)$ to $\beta = \frac{2}{x_R^2} = 2$ (obtained for large $x >0).$
• Figure 3: Graphs of $T(x,r) ,$ as functions of $r>0,$ for fixed $x_R=2$ and for the values of $x$ contained in the first column of Table \ref{['tab2']} (on the horizontal axes $r);$ the point of mimimum $r_{m}(x)$ increases from $\alpha = \frac{1}{2x_R^2} = 1/8$ (obtained at $x =0.0001)$ to $\beta = \frac{2}{x_R^2} = 1/2$ (obtained for large $x >0).$
• Figure 4: Graphs of $r_{m}(x)$ (left panel), and $m(x)= T (x,r_{m}(x))$ (right panel), as functions of $x >0,$ for fixed $x_R=1$ and for the values of $x$ contained in the first column of Table \ref{['tab2']} (on the horizontal axes $x);$$r_{m}(x)$ increases from $\alpha = 1/2$ to $\beta =2,$ while $m (x)$ increases from about $0$ to $\frac{e^ {x_R \sqrt {2\beta}}}{\beta} = \frac{1}{2} e ^ 2 \approx 3.695 \ .$
• Figure 5: Graphs of $T(x,r) ,$ as functions of $r>0,$ in the two-boundary case with $b=1$ and $x_R=0.2 \ ,$ for the values of $x$ going from $0.1$ to $0.5,$ with step $0.1$ (on the horizontal axes $r);$ the lower and upper curve correspond to $x = 0.1$ and $x = 0.5,$ respectively. As $x$ increases from $0.1$ to $0.5,$ the value $r_{m}(x)$ at which the minimum of $T(x,r)$ is attained, increases from $14.948$ to $45.009 ,$ while $m(x)= T(x, r_m(x))$ increases from $0.0804$ to $0.1451 .$

Theorems & Definitions (1)

• Remark 2.1