论文标题
在具有位置和职业时间重置的部分吸收培养基中扩散
Diffusion in a partially absorbing medium with position and occupation time resetting
论文作者
论文摘要
在本文中，我们考虑在一个$ω$中扩散，其中包含部分吸收目标$ \ calm $，并重置位置和职业时间。职业时间$ a_t $是一个布朗的功能，它决定了粒子在$ \ calm $ call $ [0，t] $上花费的时间。我们假设在时间$ t $上存在某些内部状态$ \ calu_t $，每当粒子在$ \ call $内散布时，都会对其进行修改。状态$ \ calu_t $被认为是$ a_t $的单调增加的功能，并且在$ \ calu_t $越过一些固定阈值时，吸收就会发生。我们首先展示了如何根据对$（\ x_t，a_t，a_t）$的联合概率密度或广义传播器$ p（\ x，a，t | \ x_0）$分析阈值的吸收。然后，我们引入了一个广义的随机重置协议，其中$ \ x_t $和内部状态$ \ calu_t $均已重置为其初始值，$ \ x_t \ rightArrow \ rightArrow \ rightArrow \ x_0 $和$ \ calu_t \ calu_t \ rightarrow 0 $，以Poisson Rate ry $ $ $ $ r $ $ r $ $ $ r $。后者在数学上等同于重置职业时间，$ a_t \ rightarrow 0 $。由于重置受续订过程的控制，因此可以通过生存概率表示复位的生存概率而无需重置，这意味着可以通过计算$ p（\ x，a，a，t | \ x_0）$ $ p和$ t $和$ t $和$ $ p的双拉laplace变换来确定吸收的统计数据。为了开发基本理论，我们将重点放在有限或半无限间隔给出的$ \ call $的一维（1D）扩散上，并探索重置的MFPT如何取决于各种模型参数。我们还将阈值机理与恒定吸收率的经典情况进行了比较。
In this paper we consider diffusion in a domain $Ω$ containing a partially absorbing target $\calM$ with position and occupation time resetting. The occupation time $A_t$ is a Brownian functional that determines the amount of time that the particle spends in $ \calM$ over the time interval $[0,t]$. We assume that there exists some internal state $\calU_t$ of the particle at time $t$ which is modified whenever the particle is diffusing within $\calM $. The state $\calU_t$ is taken to be a monotonically increasing function of $A_t$, and absorption occurs as soon as $\calU_t$ crosses some fixed threshold. We first show how to analyze threshold absorption in terms of the joint probability density or generalized propagator $P(\x,a,t|\x_0)$ for the pair $(\X_t,A_t)$ in the case of a non-absorbing substrate $\calM$, where $\X_t$ is the particle position at time $t$ and $\x_0$ is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position $\X_t$ and the internal state $\calU_t$ are reset to their initial values, $\X_t\rightarrow \x_0$ and $\calU_t\rightarrow 0$, at a Poisson rate $r$. The latter is mathematically equivalent to resetting the occupation time, $A_t\rightarrow 0$. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the double Laplace transform of $P(\x,a,t|\x_0)$ with respect to $t$ and $a$. In order to develop the basic theory, we focus on one-dimensional (1D) diffusion with $\calM$ given by a finite or semi-infinite interval, and explore how the MFPT with resetting depends on various model parameters. We also compare the threshold mechanism with the classical case of a constant absorption rate.
