论文标题
Brownian跳跃和Brownian Bridge重置以在线和太空中寻找高斯目标的比较
Comparison of Brownian jump and Brownian bridge resetting in search for Gaussian target on the line and in space
论文作者
论文摘要
对于$ d \ ge1 $和$ r> 0 $，令$ x^{（d; r）}（\ cdot）$为$ d $ - 维二维的布朗尼运动，带有扩散系数$ d $，配备了指数级时钟，带有费率$ r $。当时钟响起时，过程会跳到原点并重新开始。对于一个参数$ t> 0 $，令$ x^{\ text {bb}，d; t}（\ cdot）$是执行A $ D $ - 维数的brownian桥，带有扩散系数$ d $ and Bridge $ t $，然后在时间$ t $启动$ x^$ x^^d; t $ x^{d;带有扩散系数$ d $的运动直到时间$ t $，届时它将跳到原点并重新开始。用$ e_0^{d; r}表示期望，e_0^{\ text {bb}，d; t} $和$ e_0^{d; t; t} $。这些马尔可夫通过重置搜索随机目标$ a \ in \ mathbb {r}^d $与中心的高斯分布$σ^2 $，用$μ__{σ^2}^{\ text {gauss}，d} $表示。修复$ε_0> 0 $。令$τ_a$为$ a $的打击时间，以$ d = 1 $，而$ε_0$ - $ a $ a $ a $的打击时间，对于$ d \ ge2 $。在每个过程中找到目标的预期时间为$ \ int _ {\ mathbb {r}^d} \ big（e_0^*τ_a\ big）μ__{σ^2}^2}^\ text {\ text {gauss} {gauss}，d}，d}，d}，da） e_0^{\ text {bb}，d; t} $或$ e_0^{d; t; t} $。对于$ d = 1 $和$ d = 3 $，我们计算出上述每个表达式的最大值，$ r> r> 0 $或$ t> 0 $适当，以比较三个搜索过程的相对效率。就参数$ d $和$σ$而言，在1维情况下，这些Intima比例为$ \ frac {σ^2} d $，这是一种自然的缩放，但在3维情况下，它们会单位缩放为$ \ frac {σ^3} d $。我们还表明，在二维情况下，三个搜索过程中第一个的$ r> 0 $超过$ r> 0 $，如一维情况，为$ \ frac {σ^2} d $。
For $d\ge1$ 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 $σ^2$, denoted by $μ_{σ^2}^{\text{Gauss},d}$. Fix $ε_0>0$. Let $τ_a$ be the hitting time of $a$, for $d=1$, and the hitting time of the $ε_0$-ball around $a$, for $d\ge2$. The expected time to locate the target for each of the processes is $\int_{\mathbb{R}^d}\big(E_0^*τ_a\big)μ_{σ^2}^{\text{Gauss},d}(da)$, where $E_0^*$ 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 $σ$, in the 1-dimensional case these infima scale as $\frac{σ^2}D$, which is a natural scaling, but in the 3-dimensional case, they scale anomalously as $\frac{σ^3}D$. We also show that in the 2-dimensional case, the infimum over $r>0$ for the first of the three search processes scales as $\frac{σ^2}D$ as in the 1-dimensional case.
