{"title":"物理学家的西格蒙德二元性:连续时间和离散时间随机过程的空间特性与首过特性之间的桥梁","authors":"Mathis Guéneau, Léo Touzo","doi":"10.1088/1742-5468/ad6134","DOIUrl":null,"url":null,"abstract":"We consider a generic one-dimensional stochastic process <italic toggle=\"yes\">x</italic>(<italic toggle=\"yes\">t</italic>), or a random walk <italic toggle=\"yes\">X<sub>n</sub></italic>, which describes the position of a particle evolving inside an interval <inline-formula>\n<tex-math><?CDATA $[a,b]$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn1.gif\"></inline-graphic></inline-formula>, with absorbing walls located at <italic toggle=\"yes\">a</italic> and <italic toggle=\"yes\">b</italic>. In continuous time, <italic toggle=\"yes\">x</italic>(<italic toggle=\"yes\">t</italic>) is driven by some equilibrium process <inline-formula>\n<tex-math><?CDATA ${\\boldsymbol \\theta}(t)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">θ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn2.gif\"></inline-graphic></inline-formula>, while in discrete time, the jumps of <italic toggle=\"yes\">X<sub>n</sub></italic> follow a stationary process that obeys a time-reversal property. An important observable to characterize its behavior is the exit probability <inline-formula>\n<tex-math><?CDATA $E_b(x,t)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn3.gif\"></inline-graphic></inline-formula>, which is the probability for the particle to be absorbed first at the wall <italic toggle=\"yes\">b</italic>, before or at time <italic toggle=\"yes\">t</italic>, given its initial position <italic toggle=\"yes\">x</italic>. In this paper we show that the derivation of this quantity can be tackled by studying a dual process <italic toggle=\"yes\">y</italic>(<italic toggle=\"yes\">t</italic>) very similar to <italic toggle=\"yes\">x</italic>(<italic toggle=\"yes\">t</italic>) but with hard walls at <italic toggle=\"yes\">a</italic> and <italic toggle=\"yes\">b</italic>. More precisely, we show that the quantity <inline-formula>\n<tex-math><?CDATA $E_b(x,t)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn4.gif\"></inline-graphic></inline-formula> for the process <italic toggle=\"yes\">x</italic>(<italic toggle=\"yes\">t</italic>) is equal to the probability <inline-formula>\n<tex-math><?CDATA $\\tilde \\Phi(x,t|b)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Φ</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>b</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn5.gif\"></inline-graphic></inline-formula> of finding the dual process inside the interval <inline-formula>\n<tex-math><?CDATA $[a,x]$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn6.gif\"></inline-graphic></inline-formula> at time <italic toggle=\"yes\">t</italic>, with <inline-formula>\n<tex-math><?CDATA $y(0) = b$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6134ieqn7.gif\"></inline-graphic></inline-formula>. This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes that are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete- and continuous-time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes, such as fractional Brownian motion.","PeriodicalId":17207,"journal":{"name":"Journal of Statistical Mechanics: Theory and Experiment","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous- and discrete-time stochastic processes\",\"authors\":\"Mathis Guéneau, Léo Touzo\",\"doi\":\"10.1088/1742-5468/ad6134\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider a generic one-dimensional stochastic process <italic toggle=\\\"yes\\\">x</italic>(<italic toggle=\\\"yes\\\">t</italic>), or a random walk <italic toggle=\\\"yes\\\">X<sub>n</sub></italic>, which describes the position of a particle evolving inside an interval <inline-formula>\\n<tex-math><?CDATA $[a,b]$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy=\\\"false\\\">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn1.gif\\\"></inline-graphic></inline-formula>, with absorbing walls located at <italic toggle=\\\"yes\\\">a</italic> and <italic toggle=\\\"yes\\\">b</italic>. In continuous time, <italic toggle=\\\"yes\\\">x</italic>(<italic toggle=\\\"yes\\\">t</italic>) is driven by some equilibrium process <inline-formula>\\n<tex-math><?CDATA ${\\\\boldsymbol \\\\theta}(t)$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mi mathvariant=\\\"bold-italic\\\">θ</mml:mi></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn2.gif\\\"></inline-graphic></inline-formula>, while in discrete time, the jumps of <italic toggle=\\\"yes\\\">X<sub>n</sub></italic> follow a stationary process that obeys a time-reversal property. An important observable to characterize its behavior is the exit probability <inline-formula>\\n<tex-math><?CDATA $E_b(x,t)$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn3.gif\\\"></inline-graphic></inline-formula>, which is the probability for the particle to be absorbed first at the wall <italic toggle=\\\"yes\\\">b</italic>, before or at time <italic toggle=\\\"yes\\\">t</italic>, given its initial position <italic toggle=\\\"yes\\\">x</italic>. In this paper we show that the derivation of this quantity can be tackled by studying a dual process <italic toggle=\\\"yes\\\">y</italic>(<italic toggle=\\\"yes\\\">t</italic>) very similar to <italic toggle=\\\"yes\\\">x</italic>(<italic toggle=\\\"yes\\\">t</italic>) but with hard walls at <italic toggle=\\\"yes\\\">a</italic> and <italic toggle=\\\"yes\\\">b</italic>. More precisely, we show that the quantity <inline-formula>\\n<tex-math><?CDATA $E_b(x,t)$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn4.gif\\\"></inline-graphic></inline-formula> for the process <italic toggle=\\\"yes\\\">x</italic>(<italic toggle=\\\"yes\\\">t</italic>) is equal to the probability <inline-formula>\\n<tex-math><?CDATA $\\\\tilde \\\\Phi(x,t|b)$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\\\"normal\\\">Φ</mml:mi><mml:mo stretchy=\\\"true\\\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\\\"false\\\">|</mml:mo></mml:mrow><mml:mi>b</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn5.gif\\\"></inline-graphic></inline-formula> of finding the dual process inside the interval <inline-formula>\\n<tex-math><?CDATA $[a,x]$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\\\"false\\\">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn6.gif\\\"></inline-graphic></inline-formula> at time <italic toggle=\\\"yes\\\">t</italic>, with <inline-formula>\\n<tex-math><?CDATA $y(0) = b$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\\\"false\\\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\\\"jstatad6134ieqn7.gif\\\"></inline-graphic></inline-formula>. This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes that are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete- and continuous-time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes, such as fractional Brownian motion.\",\"PeriodicalId\":17207,\"journal\":{\"name\":\"Journal of Statistical Mechanics: Theory and Experiment\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Statistical Mechanics: Theory and Experiment\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1088/1742-5468/ad6134\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Mechanics: Theory and Experiment","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1742-5468/ad6134","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
摘要
我们考虑一个通用的一维随机过程 x(t)或随机游走 Xn,它描述了一个粒子在区间 [a,b] 内演化的位置,吸收墙位于 a 和 b 处。在连续时间内,x(t) 由某个均衡过程 θ(t) 驱动,而在离散时间内,Xn 的跳跃遵循一个服从时间反转特性的静止过程。描述其行为特征的一个重要观测指标是出口概率 Eb(x,t),即粒子在给定其初始位置 x 的情况下,在时间 t 之前或 t 时首先被壁 b 吸收的概率。在本文中,我们将展示如何通过研究与 x(t) 非常相似但在 a 和 b 处有硬壁的对偶过程 y(t) 来解决这个问题。更准确地说,我们将展示过程 x(t) 的 Eb(x,t) 等价于在时间 t 在区间 [a,x] 内找到对偶过程的概率 Φ~(x,t|b),其中 y(0)=b 。这在数学中被称为西格蒙德对偶性。在这里,我们将证明这种对偶性适用于物理学中各种令人感兴趣的过程,包括活动粒子模型、扩散弥散模型、一大类离散和连续时间随机漫步,甚至是受随机重置影响的过程。对于所有这些情况,我们都提供了对偶过程的明确构造。我们还给出了连续和离散时间背景下这一特性的简单推导,以及大量相关模型的数值检验。最后,我们通过模拟来证明,对于更复杂的过程,如分数布朗运动,对偶性也可能成立。
Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous- and discrete-time stochastic processes
We consider a generic one-dimensional stochastic process x(t), or a random walk Xn, which describes the position of a particle evolving inside an interval [a,b], with absorbing walls located at a and b. In continuous time, x(t) is driven by some equilibrium process θ(t), while in discrete time, the jumps of Xn follow a stationary process that obeys a time-reversal property. An important observable to characterize its behavior is the exit probability Eb(x,t), which is the probability for the particle to be absorbed first at the wall b, before or at time t, given its initial position x. In this paper we show that the derivation of this quantity can be tackled by studying a dual process y(t) very similar to x(t) but with hard walls at a and b. More precisely, we show that the quantity Eb(x,t) for the process x(t) is equal to the probability Φ~(x,t|b) of finding the dual process inside the interval [a,x] at time t, with y(0)=b. This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes that are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete- and continuous-time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes, such as fractional Brownian motion.
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