标题： 带随机重置的扩散的条件后向和前向时间：一种更新理论方法 显示英文标题

标题： Conditioned backward and forward times of diffusion with stochastic resetting: a renewal theory approach

摘要： 随机重置可以自然地被理解为一种更新过程，该过程控制着底层随机过程的演化。 在本工作中，我们从更新理论的角度正式推导了已知的带有重置的扩散结果。 与更新理论中的概念并行，我们引入了具有重置的随机过程的条件后向时间和前向时间，即在已知系统当前状态的情况下，自上次重置以来的时间和到下次重置的时间。 我们专注于研究马尔可夫和非马尔可夫重置下的扩散。 对于这些情况，我们找到了条件后向时间和前向时间的概率密度函数，并将其与过程的数值模拟进行比较。 特别是，我们发现对于具有渐近形式$\varphi(t)\sim t^{-1-\alpha}$的幂律重置时间概率密度函数，在$\alpha$的半整数值处，条件后向时间和前向时间的性质会发生显著变化。 这是由于扩散的长时间尺度$P(x,t)\sim 1/\sqrt{t}$与重置时间概率密度函数之间的组合所致。 显示英文摘要

摘要： Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward and forward times for stochastic processes with resetting to be the times since the last and until the next reset, given that the current state of the system is known. We focus on studying diffusion under Markovian and non-Markovian resetting. For these cases, we find the conditioned backward and forward time PDFs, comparing them with numerical simulations of the process. In particular, we find that for power-law reset time PDFs with asymptotic form $\varphi(t)\sim t^{-1-\alpha}$, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of $\alpha$. This is due to the composition between the long-time scaling of diffusion $P(x,t)\sim 1/\sqrt{t}$ and the reset time PDF.