Comparison of Brownian jump and Brownian bridge resetting in search for Gaussian target on the line and in space

Abstract

For $d\ge 1$ and $r>0$, let $X^{(d;r)}(\cdot )$ be a $d$-dimensional Brownian motion with diffusion coefficient $D$, equipped with an exponential clock with rate $r$. When the clock rings, the process jumps to the origin and begins anew. For a parameter $T>0$, let $X^{\text{bb},d;T}(\cdot )$ be the process that performs a $d$-dimensional Brownian bridge with diffusion coefficient $D$ and bridge interval $T$, and then at time $T$ starts anew from the origin, and let $X^{d;T}$ be the process that performs a $d$-dimensional Brownian motion with diffusion coefficient $D$ up until time $T$, at which time it jumps to the origin and begins anew. Denote expectations by $E_0^{d;r},E_0^{\text{bb},d;T}$ and $E_0^{d;T}$. These Markov processes with resetting search for a random target $a\in {\mathbb{R}}^d$ with centered Gaussian distribution of variance ${\sigma }^2$, denoted by ${\mu }_{{\sigma }^2}^{\text{Gauss},d}$. Fix ${\epsilon }_0>0$. Let ${\tau }_a$ be the hitting time of $a$, for $d=1$, and the hitting time of the ${\epsilon }_0$-ball around $a$, for $d\ge 2$. The expected time to locate the target for each of the processes is ${\int }_{{\mathbb{R}}^d}\left(E_0^{\ast }{\tau }_a\right){\mu }_{{\sigma }^2}^{\text{Gauss},d}(da)$, where $E_0^{\ast }$ stands for $E_0^{d;r},E_0^{\text{bb},d;T}$ or $E_0^{d;T}$. For $d=1$ and $d=3$, we calculate the infimum of each of the above expressions over $r>0$ or $T>0$ as appropriate, in order to compare the relative efficiencies of the three search processes. In terms of the parameters $D$ and $\sigma$, in the 1-dimensional case these infima scale as $\frac{{\sigma }^2}{D}$, which is a natural scaling, but in the 3-dimensional case, they scale anomalously as $\frac{{\sigma }^3}{D}$. We also show that in the 2-dimensional case, the infimum over $r>0$ for the first of the three search processes scales as $\frac{{\sigma }^2}{D}$ as in the 1-dimensional case.

arXiv commentMinor editing and three figures added