Relaxed Markov processes

1983 ◽  
Vol 15 (4) ◽  
pp. 769-782 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Equilibrium Distribution ◽  
Closed Model ◽  
Jackson Networks ◽  
The Creation

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.

Relaxed Markov processes

1983 ◽  
Vol 15 (04) ◽  
pp. 769-782 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Equilibrium Distribution ◽  
Closed Model ◽  
Jackson Networks ◽  
The Creation

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.

scholarly journals A Second Law for Open Markov Processes

2016 ◽  
Vol 23 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Blake S. Pollard
Keyword(s):  
Free Energy ◽  
Markov Process ◽  
Markov Processes ◽  
Relative Entropy ◽  
Upper Bound ◽  
Equilibrium Distribution ◽  
Detailed Balance ◽  
Rate Of Change ◽  
Second Law ◽  
Boundary States

In this paper we define the notion of an open Markov process. An open Markov process is a generalization of an ordinary Markov process in which populations are allowed to flow in and out of the system at certain boundary states. We show that the rate of change of relative entropy in an open Markov process is less than or equal to the flow of relative entropy through its boundary states. This can be viewed as a generalization of the Second Law for open Markov processes. In the case of a Markov process whose equilibrium obeys detailed balance, this inequality puts an upper bound on the rate of change of the free energy for any non-equilibrium distribution.

Replacement of train wheels: an application of dynamic reversal of a Markov process

1994 ◽  
Vol 31 (01) ◽  
pp. 1-8
Author(s):  
David Gates ◽  
Mark Westcott
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Time Reversal ◽  
Equilibrium Distribution ◽  
Natural Extension ◽  
Product Form ◽  
Simple Derivation ◽  
Original Process ◽  
Population Process ◽  
Ordinary Time

A problem of regrinding and recycling worn train wheels leads to a Markov population process with distinctive properties, including a product-form equilibrium distribution. A convenient framework for analyzing this process is via the notion of dynamic reversal, a natural extension of ordinary (time) reversal. The dynamically reversed process is of the same type as the original process, which allows a simple derivation of some important properties. The process seems not to belong to any class of Markov processes for which stationary distributions are known.

Replacement of train wheels: an application of dynamic reversal of a Markov process

1994 ◽  
Vol 31 (1) ◽  
pp. 1-8
Author(s):  
David Gates ◽  
Mark Westcott
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Time Reversal ◽  
Equilibrium Distribution ◽  
Natural Extension ◽  
Product Form ◽  
Simple Derivation ◽  
Original Process ◽  
Population Process ◽  
Ordinary Time

A problem of regrinding and recycling worn train wheels leads to a Markov population process with distinctive properties, including a product-form equilibrium distribution. A convenient framework for analyzing this process is via the notion of dynamic reversal, a natural extension of ordinary (time) reversal. The dynamically reversed process is of the same type as the original process, which allows a simple derivation of some important properties. The process seems not to belong to any class of Markov processes for which stationary distributions are known.

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

A temporal factorization at the maximum for certain positive self-similar Markov processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1045-1069
Author(s):  
Matija Vidmar
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Markov Processes ◽  
Absorption Time ◽  
The Laplace Transform ◽  
Self Similar

AbstractFor a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

Extension of Wald's first lemma to Markov processes

1999 ◽  
Vol 36 (01) ◽  
pp. 48-59 ◽  
Author(s):  
George V. Moustakides
Keyword(s):  
Integral Equation ◽  
Markov Process ◽  
Markov Processes ◽  
Correction Term ◽  
Stopping Time ◽  
Poisson Integral ◽  
Homogeneous Markov Process ◽  
Partial Sum ◽  
Poisson Integral Equation ◽  
Scalar Nonlinearity

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let S n denote the partial sum S n = θ(ξ1) + … + θ(ξ n ), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼S N = [lim n→∞𝔼θ(ξ n )]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξ N ). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξ N ) = o(𝔼N).

scholarly journals A Markov model for Selmer ranks in families of twists

2014 ◽  
Vol 150 (7) ◽  
pp. 1077-1106 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Elliptic Curve ◽  
Arbitrary Number ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Twists ◽  
The Family ◽  
Positive Integers

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

Composable Markov processes

1970 ◽  
Vol 7 (2) ◽  
pp. 400-410 ◽  
Author(s):  
Tore Schweder
Keyword(s):  
Social Sciences ◽  
Markov Process ◽  
Markov Processes ◽  
Infinitesimal Generator ◽  
A Priori ◽  
The Social

Many phenomena studied in the social sciences and elsewhere are complexes of more or less independent characteristics which develop simultaneously. Such phenomena may often be realistically described by time-continuous finite Markov processes. In order to define such a model which will take care of all the relevant a priori information, there ought to be a way of defining a Markov process as a vector of components representing the various characteristics constituting the phenomenon such that the dependences between the characteristics are represented by explicit requirements on the Markov process, preferably on its infinitesimal generator.

scholarly journals On invariant measures of nonlinear Markov processes

1993 ◽  
Vol 6 (4) ◽  
pp. 385-406 ◽  
Author(s):  
N. U. Ahmed ◽  
Xinhong Ding
Keyword(s):  
Markov Process ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Markov Processes ◽  
Existence And Uniqueness ◽  
Invariant Measures ◽  
Nonlinear Markov Processes

We consider a nonlinear (in the sense of McKean) Markov process described by a stochastic differential equations in Rd. We prove the existence and uniqueness of invariant measures of such process.

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