Invariant Measure for the Markov Process Corresponding to a PDE System

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

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VALIDATING THE DIFFUSION APPROXIMATION THROUGH CONDITIONAL ENTROPIES

Modern Physics Letters B ◽

10.1142/s021798491102581x ◽

2011 ◽

Vol 25 (06) ◽

pp. 377-383

Author(s):

J.-P. RIVET ◽

F. DEBBASCH

Keyword(s):

Markov Process ◽

Invariant Measure ◽

Diffusion Approximation ◽

Particle Transport ◽

Invariant Measures ◽

Large Range ◽

Brownian Particle ◽

Conditional Entropy ◽

The Real ◽

Transport Dynamics

The diffusion approximation replaces a real transport dynamics by an approximate stochastic Markov process. It is proposed that, when both dynamics have invariant measures, the conditional entropy of the invariant measure of the real dynamics with respect to the invariant measure of the Markov process be used to assess quantitatively the validity of the approximation. This proposal is tested on particle transport; the diffusion approximation is found to be quite robust, valid for an unexpectedly large range of mass ratios between the solvent and the Brownian particle.

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Existence of a σ finite invariant measure for a Markov process on a locally compact space

Israel Journal of Mathematics ◽

10.1007/bf02771598 ◽

1968 ◽

Vol 6 (1) ◽

pp. 1-4 ◽

Cited By ~ 7

Author(s):

S. R. Foguel

Keyword(s):

Markov Process ◽

Invariant Measure ◽

Compact Space ◽

Locally Compact ◽

Locally Compact Space ◽

Finite Invariant Measure

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Time Reversions of Markov Processes

Nagoya Mathematical Journal ◽

10.1017/s0027763000011405 ◽

1964 ◽

Vol 24 ◽

pp. 177-204 ◽

Cited By ~ 57

Author(s):

Masao Nagasawa

Keyword(s):

Markov Process ◽

Invariant Measure ◽

Markov Chains ◽

Markov Processes ◽

Transition Probability ◽

Initial Distribution ◽

Invariant Distribution

A time reversion of a Markov process was discussed by Kolmogoroff for Markov chains in 1936 [6] and for a diffusion in 1937 [7l He described it as a process having an adjoint transition probability. Although his treatment is purely analytical, in his case if the process xt has an invariant distribution, the reversed process zt = x-t is the process with the adjoint transition probability. In this discussion, however, it is very restrictive that the initial distribution of the process must be an invariant measure.

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Subgeometric rates of convergence for Markov processes under subordination

Advances in Applied Probability ◽

10.1017/apr.2016.83 ◽

2017 ◽

Vol 49 (1) ◽

pp. 162-181 ◽

Cited By ~ 2

Author(s):

Chang-Song Deng ◽

René L. Schilling ◽

Yan-Hong Song

Keyword(s):

Markov Process ◽

Invariant Measure ◽

Convergence Rate ◽

Rate Of Convergence ◽

Rates Of Convergence ◽

Bernstein Function ◽

Convergence To Equilibrium ◽

Speed Of Convergence ◽

Crucial Point ◽

Original Process

Abstract We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

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