Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any $t>0,$ the density (with respect to the $(d+1)$ -dimensional Lebesgue measure) of the pair $big(M_t,X_tbig)$ is a weak solution of a Fokker–Planck partial differential equation on the closed set $big{(m,x)in mathbb{R}^{d+1},,{mgeq x^1}big},$ using an integral expansion of this density.
{"title":"PDE for the joint law of the pair of a continuous diffusion and its running maximum","authors":"L. Coutin, M. Pontier","doi":"10.1017/apr.2022.76","DOIUrl":"https://doi.org/10.1017/apr.2022.76","url":null,"abstract":"\u0000\t <jats:p>Let <jats:italic>X</jats:italic> be a <jats:italic>d</jats:italic>-dimensional diffusion and <jats:italic>M</jats:italic> the running supremum of its first component. In this paper, we show that for any <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$t>0,$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> the density (with respect to the <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$(d+1)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-dimensional Lebesgue measure) of the pair <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$big(M_t,X_tbig)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is a weak solution of a Fokker–Planck partial differential equation on the closed set <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$big{(m,x)in mathbb{R}^{d+1},,{mgeq x^1}big},$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> using an integral expansion of this density.</jats:p>","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45018909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-27DOI: 10.3390/brainsci12101305
Reinhard Werth
Although subjective conscious experience and introspection have long been considered unscientific and banned from psychology, they are indispensable in scientific practice. These terms are used in scientific contexts today; however, their meaning remains vague, and earlier objections to the distinction between conscious experience and unconscious processing, remain valid. This also applies to the distinction between conscious visual perception and unconscious visual processing. Damage to the geniculo-striate pathway or the visual cortex results in a perimetrically blind visual hemifield contralateral to the damaged hemisphere. In some cases, cerebral blindness is not absolute. Patients may still be able to guess the presence, location, shape or direction of movement of a stimulus even though they report no conscious visual experience. This "unconscious" ability was termed "blindsight". The present paper demonstrates how the term conscious visual experience can be introduced in a logically precise and methodologically correct way and becomes amenable to scientific examination. The distinction between conscious experience and unconscious processing is demonstrated in the cases of conscious vision and blindsight. The literature on "blindsight" and its neurobiological basis is reviewed. It is shown that blindsight can be caused by residual functions of neural networks of the visual cortex that have survived cerebral damage, and may also be due to an extrastriate pathway via the midbrain to cortical areas such as areas V4 and MT/V5.
{"title":"A Scientific Approach to Conscious Experience, Introspection, and Unconscious Processing: Vision and Blindsight.","authors":"Reinhard Werth","doi":"10.3390/brainsci12101305","DOIUrl":"10.3390/brainsci12101305","url":null,"abstract":"<p><p>Although subjective conscious experience and introspection have long been considered unscientific and banned from psychology, they are indispensable in scientific practice. These terms are used in scientific contexts today; however, their meaning remains vague, and earlier objections to the distinction between conscious experience and unconscious processing, remain valid. This also applies to the distinction between conscious visual perception and unconscious visual processing. Damage to the geniculo-striate pathway or the visual cortex results in a perimetrically blind visual hemifield contralateral to the damaged hemisphere. In some cases, cerebral blindness is not absolute. Patients may still be able to guess the presence, location, shape or direction of movement of a stimulus even though they report no conscious visual experience. This \"unconscious\" ability was termed \"blindsight\". The present paper demonstrates how the term conscious visual experience can be introduced in a logically precise and methodologically correct way and becomes amenable to scientific examination. The distinction between conscious experience and unconscious processing is demonstrated in the cases of conscious vision and blindsight. The literature on \"blindsight\" and its neurobiological basis is reviewed. It is shown that blindsight can be caused by residual functions of neural networks of the visual cortex that have survived cerebral damage, and may also be due to an extrastriate pathway via the midbrain to cortical areas such as areas V4 and MT/V5.</p>","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"25 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2022-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9599441/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90976721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wasiur R. KhudaBukhsh, Casper Woroszyło, G. Rempała, H. Koeppl
Abstract We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.
{"title":"A functional central limit theorem for SI processes on configuration model graphs","authors":"Wasiur R. KhudaBukhsh, Casper Woroszyło, G. Rempała, H. Koeppl","doi":"10.1017/apr.2022.52","DOIUrl":"https://doi.org/10.1017/apr.2022.52","url":null,"abstract":"Abstract We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"54 1","pages":"880 - 912"},"PeriodicalIF":1.2,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42366140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider a sequence of Poisson cluster point processes on $mathbb{R}^d$ : at step $ninmathbb{N}_0$ of the construction, the cluster centers have intensity $c/(n+1)$ for some $c>0$ , and each cluster consists of the particles of a branching random walk up to generation n—generated by a point process with mean 1. We show that this ‘critical cluster cascade’ converges weakly, and that either the limit point process equals the void process (extinction), or it has the same intensity c as the critical cluster cascade (persistence). We obtain persistence if and only if the Palm version of the outgrown critical branching random walk is locally almost surely finite. This result allows us to give numerous examples for persistent critical cluster cascades.
{"title":"Critical cluster cascades","authors":"Matthias Kirchner","doi":"10.1017/apr.2022.26","DOIUrl":"https://doi.org/10.1017/apr.2022.26","url":null,"abstract":"Abstract We consider a sequence of Poisson cluster point processes on \u0000$mathbb{R}^d$\u0000 : at step \u0000$ninmathbb{N}_0$\u0000 of the construction, the cluster centers have intensity \u0000$c/(n+1)$\u0000 for some \u0000$c>0$\u0000 , and each cluster consists of the particles of a branching random walk up to generation n—generated by a point process with mean 1. We show that this ‘critical cluster cascade’ converges weakly, and that either the limit point process equals the void process (extinction), or it has the same intensity c as the critical cluster cascade (persistence). We obtain persistence if and only if the Palm version of the outgrown critical branching random walk is locally almost surely finite. This result allows us to give numerous examples for persistent critical cluster cascades.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"357 - 381"},"PeriodicalIF":1.2,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47415286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We show that load-sharing models (a very special class of multivariate probability models for nonnegative random variables) can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes. Such results can be applied in several different fields. In particular, applications of them can be developed in the context of paradoxes which arise in voting theory. Also, an application to the notion of probability signature may be of interest, in the field of systems reliability.
{"title":"Construction of aggregation paradoxes through load-sharing models","authors":"Emilio De Santis, F. Spizzichino","doi":"10.1017/apr.2022.17","DOIUrl":"https://doi.org/10.1017/apr.2022.17","url":null,"abstract":"Abstract We show that load-sharing models (a very special class of multivariate probability models for nonnegative random variables) can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes. Such results can be applied in several different fields. In particular, applications of them can be developed in the context of paradoxes which arise in voting theory. Also, an application to the notion of probability signature may be of interest, in the field of systems reliability.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"223 - 244"},"PeriodicalIF":1.2,"publicationDate":"2022-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47547784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the following migration process based on a closed network of N queues with $K_N$ customers. Each station is a $cdot$ /M/ $infty$ queue with service (or migration) rate $mu$ . Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate $alpha Y$ if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate $beta$ . We let N tend to infinity and assume that $lim_{Nto infty} K_N/N= eta $ , where $eta$ is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to $eta$ . The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.
考虑以下迁移过程，该过程基于具有$K_N$客户的N个队列的封闭网络。每个站点都是一个服务(或迁移)速率为$mu$的$cdot$ /M/ $infty$队列。在出发时，客户被独立地、均匀地随机路由到另一个站点。除了迁移之外，这些客户还受到易受感染-易受感染(SIS)的影响。也就是说，客户处于两种状态之一:I代表受感染，S代表易感。客户可以在站点中从I切换到S或从S切换到I。更准确地说，在任何站点，如果站点中有Y个受感染的客户，则每个易感客户以瞬时速率$alpha Y$感染，而每个受感染的客户恢复并以速率$beta$感染。我们让N趋于无穷，并假设$lim_{Nto infty} K_N/N= eta $，其中$eta$是代表客户密度的正常数。我们感兴趣的主要问题涉及这样一个系统的一组参数，在这个系统中存在一个固定的状态，在这个状态下，流行病在极限系统中存活。后一种极限称为热力学极限。我们使用耦合和随机单调性参数来建立相关马尔可夫过程的关键属性，这反过来又允许我们给出该热力学极限相对于$eta$的相变图的结构。对这个SIS模型的Kolmogorov方程的分析简化为我们没有找到显式解的波型偏微分方程的分析。这个简单的SIS模型是表现出随机迁移和传染的几个伴生随机过程之一。本文讨论了其中的两个，因为它们提供了普通SIS模型的变体以及一些边界和近似值。这两种变体是状态变化偏离(DOCS)模型和平均感染率(AIR)模型，它们都有封闭形式的解。AIR系统是一个经典的平均场模型，其中基于实际感染客户数量的感染机制被基于所有站点感染客户数量的经验平均值的机制所取代。后者允许产品形式的解决方案。DOCS的特点是加速迁移，因为每次更改SIS状态都意味着立即离开。该模型导致了另一种允许封闭形式解的波型PDE。在本文中，主要关注的是封闭随机网络及其限制。由泊松输入的单站组成的开放系统有助于热力学极限的分析，也具有独立的意义。这类SIS动力学在几乎所有文献中的排队网络中都有体现。
{"title":"Migration–contagion processes","authors":"F. Baccelli, S. Foss, V. Shneer","doi":"10.1017/apr.2023.17","DOIUrl":"https://doi.org/10.1017/apr.2023.17","url":null,"abstract":"\u0000 Consider the following migration process based on a closed network of N queues with \u0000 \u0000 \u0000 \u0000$K_N$\u0000\u0000 \u0000 customers. Each station is a \u0000 \u0000 \u0000 \u0000$cdot$\u0000\u0000 \u0000 /M/\u0000 \u0000 \u0000 \u0000$infty$\u0000\u0000 \u0000 queue with service (or migration) rate \u0000 \u0000 \u0000 \u0000$mu$\u0000\u0000 \u0000 . Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate \u0000 \u0000 \u0000 \u0000$alpha Y$\u0000\u0000 \u0000 if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate \u0000 \u0000 \u0000 \u0000$beta$\u0000\u0000 \u0000 . We let N tend to infinity and assume that \u0000 \u0000 \u0000 \u0000$lim_{Nto infty} K_N/N= eta $\u0000\u0000 \u0000 , where \u0000 \u0000 \u0000 \u0000$eta$\u0000\u0000 \u0000 is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to \u0000 \u0000 \u0000 \u0000$eta$\u0000\u0000 \u0000 . The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47376332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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