Lumpability and Commutativity of Markov Processes

Tian, Jianjun Paul; Kannan, D.

doi:10.1080/07362990600632045

Cited by 29 publications

(27 citation statements)

References 3 publications

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“…The following lemma guarantees M(t) is lumpable. Let the matrices U and V be matrices for the partition of the state space as in Tian and Kannan (2006). We first have the following lemma, which can be proved by a straightforward computation of products of block matrices.…”

Section: Parity Lumping Of the Coalescent Process M(t)mentioning

confidence: 99%

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The Mutation Process in Colored Coalescent Theory

Tian

Lin

2009

Bull. Math. Biol.

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The mutation process is introduced into the colored coalescent theory. The mutation process can be viewed as an independent Poisson process running on the colored genealogical random tree generated by the colored coalescent process, with the edge lengths of the random tree serving as the time scale for the mutation process. Moving backward along the colored genealogical tree, the color of vertices may change in two ways, when two vertices coalesce, or when a mutation happens. The rule that governs the coalescent change of color involves a parameter x; the rule that governs the mutation involves a parameter micro. Explicit computations of the expectation of the coalescent time (the first hitting time), and the coalescent probabilities (the first hitting probabilities) are carried out. For example, our calculation shows that when x = 1/2, for a sample of n colored individuals, the expected time for the colored coalescent process with the mutation process superimposed to first reach a black MRCA or a white MRCA, respectively, is 3 -- 2/n with probability 1/2 for any value of the parameter micro. On the other hand, the expected time for the colored coalescent process with mutation to first reach a MRCA, either black or white, is 2 -- 2/n for any values of the parameters micro and x, which is the same as that for the standard Kingman coalescent process.

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Section: Parity Lumping Of the Coalescent Process M(t)mentioning

confidence: 99%

“…The technique of lumping turns out to be very important in simplifying computations involved. The lumping technique can be found in Tian and Kannan (2006).…”

Section: Introductionmentioning

confidence: 99%

The Mutation Process in Colored Coalescent Theory

Tian

Lin

2009

Bull. Math. Biol.

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“…Let us come back to (22). Since v = (p −q)/E(J 1 ) we can write (recall (39) and Corollary 6.3) (p +q)/E(J 1 ) = v(1 + ∆)/(1 − ∆) = (1 + ∆)/ N n=1 r n .…”

Section: Examples: N -Periodic Linear Modelmentioning

confidence: 99%

“…Since v = (p −q)/E(J 1 ) we can write (recall (39) and Corollary 6.3) (p +q)/E(J 1 ) = v(1 + ∆)/(1 − ∆) = (1 + ∆)/ N n=1 r n . In conclusion, plugging this last relation, (47) and (51) into (22), and recalling (42), we end up with the following expression for the diffusion coefficient in the N -periodic model:…”

Section: Examples: N -Periodic Linear Modelmentioning

confidence: 99%

Discrete Kinetic Models for Molecular Motors: Asymptotic Velocity and Gaussian Fluctuations

Faggionato

Silvestri²

2014

J Stat Phys

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Abstract. We consider random walks on quasi one dimensional lattices, as introduced in [6]. This mathematical setting covers a large class of discrete kinetic models for noncooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.

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“…In a rare event problem , large state space has been partitioned into three sets: "good states", "failed or bad states", "internal states" when he is constructing Markov model. Ridder [9] interested in first passage time probability that Markov chain will pass the failure set before the good set when the chain starts in internal state Tian and Kannan [10] …”

Section: Distribution Of First Passage Times For Lumped States In Marmentioning

confidence: 99%

Distribution of First Passage Times for Lumped States in Markov Chains

Gül¹,

Çelebioğlu²

2015

JMSS

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First passage time in Markov chains is defined as the first time that a chain passes a specified state or lumped states. This state or lumped states may indicate first passage time of an interesting, rare and amazing event. In this study, obtaining distribution of the first passage time relating to lumped states which are constructed by gathering the states through lumping method for a irreducible Markov chain whose state space is finite was deliberated. Thanks to lumping method the chain's Markov property has been preserved. Another benefit of lumping method in the way of practice is reduction of the state space thanks to gathering states together. As the obtained first passage distributions are continuous, it may be used in many fields such as reliability and risk analysis

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Lumpability and Commutativity of Markov Processes

Cited by 29 publications

References 3 publications

The Mutation Process in Colored Coalescent Theory

The Mutation Process in Colored Coalescent Theory

Discrete Kinetic Models for Molecular Motors: Asymptotic Velocity and Gaussian Fluctuations

Distribution of First Passage Times for Lumped States in Markov Chains

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