Naturality, standardness, and weak duality for Markov processes

One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which 3 disjoint infinite clusters are present.

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“…Fatou's lemma yields E |T | < ∞ and hence P |T | < ∞ = 1. We finally conclude that P[T = ∅] = 0 by combining [11,Theorem 6.7] and [10, (2.9)]. …”

Section: Upper Bounds For K-arm Timesmentioning

confidence: 78%

Quantitative noise sensitivity and exceptional times for percolation

2010

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“…We can (and do) assume that a version of I has been chosen such that 0 < / < 1. Arguing as in [29] one sees that 1r/?I < U < l[a,/3[ (modulo Im).…”

Section: Preliminariesmentioning

confidence: 92%

“…Such matters (relative to ç in the context of right processes) have been clarified by Getoor and Sharpe in [29] and we follow their lead in defining a process / by (3-3) l = p(h«M)-Set A = {/ > 0} = {(î,w) E R x W: lt(w) > 0}. We can (and do) assume that a version of I has been chosen such that 0 < / < 1.…”

Section: Preliminariesmentioning

confidence: 98%

“…Recall from [29] that a Borel set B E E is m-polar provided Qm(Yt E B, some i E R) = 0. Evidently B E £ is m-polar if and only if Pm(Xs E B, some s > 0) = 0.…”

Section: Jnmentioning

confidence: 99%

“…The process (Yt, Qm) has proved to be quite useful in studying certain aspects of the process (Xt,Px). See, for example, [1,14,15,16,29,30,36,38,39]. In particular, (Yt,Qm) was used to study the convex cone Exc in [16,19,30].…”

Section: Introductionmentioning

confidence: 99%

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Homogeneous random measures and a weak order for the excessive measures of a Markov process

Fitzsimmons¹

1987

Trans. Amer. Math. Soc.

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ABSTRACT. Let X = (Xt,Px) be a right Markov process and let m be an excessive measure for X. Associated with the pair (X, ra) is a stationary strong Markov process (Yt, Qm) with random times of birth and death, with the same transition function as X, and with m as one dimensional distribution.We use (Yt,Qm) to study the cone of excessive measures for X. A "weak order" is defined on this cone: an excessive measure £ is weakly dominated by m if and only if there is a suitable homogeneous random measure re such that (Yt,Qç) is obtained by "birthing" (Yt,Qm), birth in [t,t + dt] occurring at rate re(di). Random measures such as re are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over (Yt,Qm), including the moderate Markov property of {Yt,Qm) when the arrow of time is reversed. Applications to balayage and capacity are suggested. Introduction.Let (Ps : s > 0) be a Borel right semigroup on a Lusin state space (E, £) and let (Xt, Px) denote the associated right continuous strong Markov process. Recall that a measure m on (E, £) is excessive for (Ps) if m is tr-finite and if mPs < m for all s > 0. Since (Ps) is a right semigroup, one then has mPs ] m as s J. 0, as is well known. Let Exc denote the convex cone of excessive measures for (Pa)-Given m E Exc, according to a theorem of Kuznetsov [31], one can construct a stationary process (Yt : t E R) having random birth and death times and a tr-finite governing measure Qm, such that for ii < t2 < • ■ ■ < tn (i» E R)

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Reversibility in Queueing Models

Miyazawa

2011

Wiley Encyclopedia of Operations Research and Management Science

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In stochastic models for queues and their networks, random events evolve in time. A process for their backward evolution is referred to as a time‐reversed process . If some property is unchanged under time reversal, we may better understand the model. A concept of reversibility is invented for this invariance. Local balance for a stationary Markov chain has been used, but it is still too strong for queueing applications. We are concerned with a continuous‐time Markov chain, but do not assume it has a stationary distribution. We define reversibility in structure as an invariant property of a family of the set of models under a certain operation. Any member of this set is a pair of transition rate functions and its supporting measure, and each set represents dynamics of queueing systems such as arrivals and departures. We use a permutation Γ of the family members to describe the change of the dynamics under time reversal. This reversibility is called Γ ‐reversibility in structure . To apply these definitions, we introduce new classes of models, called reacting systems and self‐reacting systems . Using them, we give a unified view for queues and their networks. They include symmetric service, batch movements, and state‐dependent routing.

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Naturality, standardness, and weak duality for Markov processes

Cited by 98 publications

References 31 publications

Quantitative noise sensitivity and exceptional times for percolation

Quantitative noise sensitivity and exceptional times for percolation

Homogeneous random measures and a weak order for the excessive measures of a Markov process

Reversibility in Queueing Models

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